Section 2.5

Integration of Miscellaneous Functions

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Subsection 2.5.2Problems[234/100%]
Problem 2.5.1

Find the integral. ∫x3−x2x+3dx\int \frac{x^{3}-x^{2}}{x+3} d x.

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Problem 2.5.2

Find the integral. ∫xdx2x2−x+1\int \frac{x d x}{2 x^{2}-x+1}.

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Problem 2.5.3

Find the integral. ∫x2dxx2+12x+35\int \frac{x^{2} d x}{x^{2}+12 x+35}.

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Problem 2.5.4

Find the integral. ∫x4+3x3−1x2+2x+1dx\int \frac{x^{4}+3 x^{3}-1}{x^{2}+2 x+1} d x.

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Problem 2.5.5

Find the integral. ∫x4dxx2+4\int \frac{x^{4} d x}{x^{2}+4}.

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Problem 2.5.6

Find the integral. ∫xdxx3+8\int \frac{x d x}{x^{3}+8}.

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Problem 2.5.7

Find the integral. ∫dx(x+1)(9x2+6x+4)\int \frac{d x}{(x+1)\left(9 x^{2}+6 x+4\right)}.

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Problem 2.5.8

Find the integral. ∫(5−x)dx(x−3)(x2−5x+7)\int \frac{(5-x) d x}{(x-3)\left(x^{2}-5 x+7\right)}.

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Problem 2.5.9

Find the integral. ∫x2−2x−5x3−x2+2x−2dx\int \frac{x^{2}-2 x-5}{x^{3}-x^{2}+2 x-2} d x.

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Problem 2.5.10

Find the integral. ∫x3dxx3−3x+2\int \frac{x^{3} d x}{x^{3}-3 x+2}.

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Problem 2.5.11

Find the integral. ∫(x3+x2)dx(x−2)(2x2−x+2)\int \frac{\left(x^{3}+x^{2}\right) d x}{(x-2)\left(2 x^{2}-x+2\right)}.

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Problem 2.5.12

Find the integral. ∫2x4−2x3−x2+22x3−4x2+3x−1dx\int \frac{2 x^{4}-2 x^{3}-x^{2}+2}{2 x^{3}-4 x^{2}+3 x-1} d x.

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Problem 2.5.13

Find the integral. ∫dxx4−x3\int \frac{d x}{x^{4}-x^{3}}.

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Problem 2.5.14

Find the integral. ∫(2x2−5)dxx4−5x2+6\int \frac{\left(2 x^{2}-5\right) d x}{x^{4}-5 x^{2}+6}.

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Problem 2.5.15

Find the integral. ∫x2dx(x2+12x+35)2\int \frac{x^{2} d x}{\left(x^{2}+12 x+35\right)^{2}}.

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Problem 2.5.16

Find the integral. ∫dxx4+8x\int \frac{d x}{x^{4}+8 x}.

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Problem 2.5.17

Find the integral. ∫(2−x)dxx4−x3−x+1\int \frac{(2-x) d x}{x^{4}-x^{3}-x+1}.

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Problem 2.5.18

Find the integral. ∫10−5x+3x2(x−1)2(x2+2x+5)dx\int \frac{10-5 x+3 x^{2}}{(x-1)^{2}\left(x^{2}+2 x+5\right)} d x.

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Problem 2.5.19

Find the integral. ∫(x2x2+4)2dx\int \left(\frac{x^{2}}{x^{2}+4}\right)^{2} d x.

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Problem 2.5.20

Find the integral. ∫19x3+9x2+6x+19x4+37x2+4dx\int \frac{19 x^{3}+9 x^{2}+6 x+1}{9 x^{4}+37 x^{2}+4} d x.

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Problem 2.5.21

Find the integral. ∫dxx4+x2+1\int \frac{d x}{x^{4}+x^{2}+1}.

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Problem 2.5.22

Find the integral. ∫(x2−1)dxx4+3x3+5x2+3x+1\int \frac{\left(x^{2}-1\right) d x}{x^{4}+3 x^{3}+5 x^{2}+3 x+1}.

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Problem 2.5.23

Find the integral. ∫x2+5x+4x4+5x2+4dx\int \frac{x^{2}+5 x+4}{x^{4}+5 x^{2}+4} d x.

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Problem 2.5.24

Find the integral. ∫1+2x+x2+4x31+x2+x4dx\int \frac{1+2 x+x^{2}+4 x^{3}}{1+x^{2}+x^{4}} d x.

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Problem 2.5.25

Find the integral. ∫5x7−5x2−18xx5+3x2−1dx\int \frac{5 x^{7}-5 x^{2}-18 x}{x^{5}+3 x^{2}-1} d x.

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Problem 2.5.26

Find the integral. ∫dx(x+1)(x2+x+1)2\int \frac{d x}{(x+1)\left(x^{2}+x+1\right)^{2}}.

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Problem 2.5.27

Find the integral. ∫x5dxx6+9x3+8\int \frac{x^{5} d x}{x^{6}+9 x^{3}+8}.

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Problem 2.5.28

Find the integral. ∫dx(x2+x+1)3\int \frac{d x}{\left(x^{2}+x+1\right)^{3}}

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Problem 2.5.29

Find the integral. ∫dxx8+7x\int \frac{d x}{x^{8}+7 x}.

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Problem 2.5.30

Find the integral. ∫dxx8+8x6+16x4\int \frac{d x}{x^{8}+8 x^{6}+16 x^{4}}.

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Problem 2.5.31

Find the integral. ∫dxx8+x4+1\int \frac{d x}{x^{8}+x^{4}+1}.

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Problem 2.5.32

Find the integral. ∫dx(x4−1)3\int \frac{d x}{\left(x^{4}-1\right)^{3}}.

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Problem 2.5.33

Find the integral. ∫dxx(3+x6)2\int \frac{d x}{x\left(3+x^{6}\right)^{2}}.

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Problem 2.5.34

Find the integral. ∫x3dx(x8+1)2\int \frac{x^{3} d x}{\left(x^{8}+1\right)^{2}}.

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Problem 2.5.35

Find the integral. ∫x9dx(x10+2x5+2)2\int \frac{x^{9} d x}{\left(x^{10}+2 x^{5}+2\right)^{2}}.

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Problem 2.5.36

Find the integral. ∫dxx+x3\int \frac{d x}{\sqrt{x}+\sqrt[3]{x}}.

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Problem 2.5.37

Find the integral. ∫xdxx+xx4\int \frac{x d x}{\sqrt{x}+x \sqrt[4]{x}}.

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Problem 2.5.38

Find the integral. ∫dxx56+4x\int \frac{d x}{\sqrt[6]{x^{5}}+4 \sqrt{x}}.

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Problem 2.5.39

Find the integral. ∫dxx+x3+x5.\int \frac{d x}{\sqrt{x}+\sqrt{x^{3}}+\sqrt{x^{5}}}.

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Problem 2.5.40

Find the integral. ∫x+x+x23x+x43dx\int \frac{x+\sqrt{x}+\sqrt[3]{x^{2}}}{x+\sqrt[3]{x^{4}}} d x

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Problem 2.5.41

Find the integral. ∫2+x+1(x+1)2−x+1dx\int \frac{2+\sqrt{x+1}}{(x+1)^{2}-\sqrt{x+1}} d x.

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Problem 2.5.42

Find the integral. ∫1−1+x61+x+(1+x)43dx\int \frac{1-\sqrt[6]{1+x}}{1+x+\sqrt[3]{(1+x)^{4}}} d x.

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Problem 2.5.43

Find the integral. ∫dx2x−1−2x−14.\int \frac{d x}{\sqrt{2 x-1}-\sqrt[4]{2 x-1}}.

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Problem 2.5.44

Find the integral. ∫1−x1+xdx\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x.

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Problem 2.5.45

Find the integral. ∫dx2+1+x+1−x\int \frac{d x}{\sqrt{2}+\sqrt{1+x}+\sqrt{1-x}}.

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Problem 2.5.46

Find the integral. ∫dxx1−x2\int \frac{d x}{x \sqrt{1-x^{2}}}.

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Problem 2.5.47

Find the integral. ∫dxx2x2−1\int \frac{d x}{x^{2} \sqrt{x^{2}-1}}

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Problem 2.5.48

Find the integral. ∫dxx31−x2\int \frac{d x}{x^{3} \sqrt{1-x^{2}}}.

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Problem 2.5.49

Find the integral. ∫x31−x2dx\int x^{3} \sqrt{1-x^{2}} d x.

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Problem 2.5.50

Find the integral. ∫x44−x2dx\int x^{4} \sqrt{4-x^{2}} d x.

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Problem 2.5.51

Find the integral. ∫dx(1−x)3+2x−x2\int \frac{d x}{(1-x) \sqrt{3+2 x-x^{2}}}.

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Problem 2.5.52

Find the integral. ∫dx(x−1)3x2−2x−1\int \frac{d x}{(x-1)^{3} \sqrt{x^{2}-2 x-1}}.

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Problem 2.5.53

Find the integral. ∫dx(x+1)5x2+2x\int \frac{d x}{(x+1)^{5} \sqrt{x^{2}+2 x}}.

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Problem 2.5.54

Find the integral. ∫xdx(x2−3x+2)x2−4x+3,x>3\int \frac{x d x}{\left(x^{2}-3 x+2\right) \sqrt{x^{2}-4 x+3}}, x > 3.

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Problem 2.5.55

Find the integral. ∫x3dx1+x2\int \frac{x^{3} d x}{\sqrt{1+x^{2}}}.

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Problem 2.5.56

Find the integral. ∫dxx41+x2\int \frac{d x}{x^{4} \sqrt{1+x^{2}}}.

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Problem 2.5.57

Find the integral. ∫dx(1−x4)1+x2\int \frac{d x}{\left(1-x^{4}\right) \sqrt{1+x^{2}}}.

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Problem 2.5.58

Find the integral. ∫x3+2x2+3x+4x2+2x+2dx\int \frac{x^{3}+2 x^{2}+3 x+4}{\sqrt{x^{2}+2 x+2}} d x.

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Problem 2.5.59

Find the integral. ∫4x2−4x+3dx\int \sqrt{4 x^{2}-4 x+3} d x.

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Problem 2.5.60

Find the integral. ∫1−4x−x2dx\int \sqrt{1-4 x-x^{2}} d x.

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Problem 2.5.61

Find the integral. ∫x2+x+1(x2+1)x2+1 dx\int \frac{x^{2}+x+1}{\left(x^{2}+1\right) \sqrt{x^{2}+1}} \, d x.

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Problem 2.5.62

Find the integral. ∫dx(7x−x2−10)3\int \frac{d x}{\sqrt{\left(7 x-x^{2}-10\right)^{3}}}.

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Problem 2.5.63

Find the integral. ∫xdx(x2+x+2)4x2+4x+3\int \frac{x d x}{\left(x^{2}+x+2\right) \sqrt{4 x^{2}+4 x+3}}.

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Problem 2.5.64

Find the integral. ∫dx(1−1−x2)2\int \frac{d x}{\left(1-\sqrt{1-x^{2}}\right)^{2}}.

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Problem 2.5.65

Find the integral. ∫dxx+x2+x+1\int \frac{d x}{x+\sqrt{x^{2}+x+1}}.

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Problem 2.5.66

Find the integral. ∫x2 dx1+2(x+x2+x+1)\int \frac{x^{2} \, d x}{1+2\left(x+\sqrt{x^{2}+x+1}\right)}.

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Problem 2.5.67

Find the integral. ∫(x+1+x2)121+x2dx\int \frac{\left(x+\sqrt{1+x^{2}}\right)^{12}}{\sqrt{1+x^{2}}} d x.

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Problem 2.5.68

Find the integral. ∫dxxx2+13\int \frac{d x}{x \sqrt[3]{x^{2}+1}}.

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Problem 2.5.69

Find the integral. ∫x5(1+x3)23dx\int x^{5} \sqrt[3]{\left(1+x^{3}\right)^{2}} d x.

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Problem 2.5.70

Find the integral. ∫1+x33x2dx\int \frac{\sqrt[3]{1+x^{3}}}{x^{2}} d x.

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Problem 2.5.71

Find the integral. ∫dxx111+x4\int \frac{d x}{x^{11} \sqrt{1+x^{4}}}.

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Problem 2.5.72

Find the integral. ∫(x2−1) dxx1+3x2+x4\int \frac{\left(x^{2}-1\right) \, d x}{x \sqrt{1+3 x^{2}+x^{4}}}.

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Problem 2.5.73

Find the integral. ∫1−x4x5dx\int \frac{\sqrt{1-x^{4}}}{x^{5}} d x.

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Problem 2.5.74

Find the integral. ∫dx(x−1)3(x−2)\int \frac{d x}{\sqrt{(x-1)^{3}(x-2)}}.

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Problem 2.5.75

Find the integral. ∫dxx3−x2−x+13\int \frac{d x}{\sqrt[3]{x^{3}-x^{2}-x+1}}.

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Problem 2.5.76

Find the integral. ∫(1+x2)dx(1−x2)1+x4\int \frac{\left(1+x^{2}\right) d x}{\left(1-x^{2}\right) \sqrt{1+x^{4}}}.

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Problem 2.5.77

Find the integral. ∫1+x4x4−1dx\int \frac{\sqrt{1+x^{4}}}{x^{4}-1} d x.

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Problem 2.5.78

Find the integral. ∫(x2−1)dxx62x2−x4−1\int \frac{\left(x^{2}-1\right) d x}{x \sqrt{62 x^{2}-x^{4}-1}}.

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Problem 2.5.79

Find the integral. ∫dx(3+5x3)3+4x33\int \frac{d x}{\left(3+5 x^{3}\right) \sqrt[3]{3+4 x^{3}}}.

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Problem 2.5.80

Find the integral. ∫dx(1+x4)1+2x44\int \frac{d x}{\left(1+x^{4}\right) \sqrt[4]{1+2 x^{4}}}.

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Problem 2.5.81

Find the integral. ∫sin⁡2xsin⁡5xdx\int \sin 2 x \sin 5 x d x.

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Problem 2.5.82

Find the integral. ∫cos⁡xcos⁡2xcos⁡3xdx\int \cos x \cos 2 x \cos 3 x d x.

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Problem 2.5.83

Find the integral. ∫sh⁡xsh⁡2xsh⁡3xdx\int \operatorname {sh} x \operatorname {sh} 2 x \operatorname {sh} 3 x d x.

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Problem 2.5.84

Find the integral. ∫dx1−sin⁡x\int \frac{d x}{1-\sin x}.

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Problem 2.5.85

Find the integral. ∫1−sin⁡xcos⁡xdx\int \frac{1-\sin x}{\cos x} d x.

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Problem 2.5.86

Find the integral. ∫sin⁡x1+sin⁡xdx\int \frac{\sin x}{1+\sin x} d x.

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Problem 2.5.87

Find the integral. ∫sin⁡2xcos⁡23xdx\int \sin^{2} x \cos^{2} 3 x d x.

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Problem 2.5.88

Find the integral. ∫sin⁡26xsin⁡23xdx\int \sin^{2} 6 x \sin^{2} 3 x d x.

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Problem 2.5.89

Find the integral. ∫cos⁡3xsin⁡5xdx\int \cos^{3} x \sin^{5} x d x.

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Problem 2.5.90

Find the integral. ∫sin⁡4xcos⁡5xdx\int \sin^{4} x \cos^{5} x d x.

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Problem 2.5.91

Find the integral. ∫sh⁡4xch⁡4xdx\int \operatorname {sh}^{4} x \operatorname {ch}^{4} x d x.

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Problem 2.5.92

Find the integral. ∫sh⁡3xch⁡4xdx\int \operatorname {sh}^{3} x \operatorname {ch}^{4} x d x.

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Problem 2.5.93

Find the integral. ∫cos⁡3xsin⁡4xdx\int \frac{\cos^{3} x}{\sin^{4} x} d x.

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Problem 2.5.94

Find the integral. ∫cos⁡5xsin⁡2xdx\int \frac{\cos^{5} x}{\sin^{2} x} d x.

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Problem 2.5.95

Find the integral. ∫ch⁡4xsh⁡2xdx\int \frac{\operatorname {ch}^{4} x}{\operatorname {sh}^{2} x} d x.

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Problem 2.5.96

Find the integral. ∫dxsh⁡2xch⁡3x\int \frac{d x}{\operatorname {sh}^{2} x \operatorname {ch}^{3} x}.

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Problem 2.5.97

Find the integral. ∫(tg⁡2x+tg⁡4x)dx\int \left(\operatorname {tg}^{2} x+\operatorname {tg}^{4} x\right) d x.

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Problem 2.5.98

Find the integral. ∫ctg⁡6xdx\int \operatorname {ctg}^{6} x d x.

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Problem 2.5.99

Find the integral. ∫tg⁡7xdx\int \operatorname {tg}^{7} x d x.

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Problem 2.5.100

Find the integral. ∫cth⁡4xdx\int \operatorname {cth}^{4} x d x.

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Problem 2.5.101

Find the integral. ∫cos⁡2xdxsin⁡xcos⁡3x\int \frac{\cos^{2} x d x}{\sin x \cos 3 x}.

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Problem 2.5.102

Find the integral. ∫cos⁡3xcos⁡4xdx\int \frac{\cos 3 x}{\cos^{4} x} d x.

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Problem 2.5.103

Find the integral. ∫sh⁡3xch⁡6xdx\int \frac{\operatorname {sh} 3 x}{\operatorname {ch}^{6} x} d x.

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Problem 2.5.104

Find the integral. ∫ch⁡3xsh⁡3xdx\int \frac{\operatorname {ch} 3 x}{\operatorname {sh}^{3} x} d x.

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Problem 2.5.105

Find the integral. ∫3cos⁡x+7sin⁡x5cos⁡x+2sin⁡xdx\int \frac{3 \cos x+7 \sin x}{5 \cos x+2 \sin x} d x.

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Problem 2.5.106

Find the integral. ∫ch⁡x4ch⁡x−3sh⁡xdx\int \frac{\operatorname {ch} x}{4 \operatorname {ch} x-3 \operatorname {sh} x} d x.

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Problem 2.5.107

Find the integral. ∫dx2+sin⁡2x+cos⁡2x\int \frac{d x}{2+\sin 2 x+\cos 2 x}.

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Problem 2.5.108

Find the integral. ∫2+cos⁡4x5+4cos⁡4xdx\int \frac{2+\cos 4 x}{5+4 \cos 4 x} d x.

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Problem 2.5.109

Find the integral. ∫3sin⁡x+2cos⁡x+1sin⁡x+3sin⁡2xdx\int \frac{3 \sin x+2 \cos x+1}{\sin x+3 \sin^{2} x} d x.

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Problem 2.5.110

Find the integral. ∫3sin⁡x+2cos⁡x+1sin⁡x+sin⁡2xdx\int \frac{3 \sin x+2 \cos x+1}{\sin x+\sin 2 x} d x.

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Problem 2.5.111

Find the integral. ∫(1+cos⁡x)21+sin⁡xdx\int \frac{(1+\cos x)^{2}}{1+\sin x} d x.

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Problem 2.5.112

Find the integral. ∫(sin⁡x−cos⁡x)sin⁡2xsin⁡3x+cos⁡3xdx\int \frac{(\sin x-\cos x) \sin 2 x}{\sin^{3} x+\cos^{3} x} d x.

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Problem 2.5.113

Find the integral. ∫3sh⁡x+2ch⁡x+1sh⁡2x+ch⁡xdx\int \frac{3 \operatorname {sh} x+2 \operatorname {ch} x+1}{\operatorname {sh} 2 x+\operatorname {ch} x} d x.

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Problem 2.5.114

Find the integral. ∫sh⁡2xdx2+sh⁡2x\int \frac{\operatorname {sh} 2 x d x}{2+\operatorname {sh}^{2} x}.

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Problem 2.5.115

Find the integral. ∫sin⁡xsin⁡2xsin⁡x+cos⁡xdx\int \frac{\sin x \sin 2 x}{\sin x+\cos x} d x.

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Problem 2.5.116

Find the integral. ∫dx4+tg⁡x+4ctg⁡x\int \frac{d x}{4+\operatorname {tg} x+4 \operatorname {ctg} x}.

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Problem 2.5.117

Find the integral. ∫ctg⁡3x+ctg⁡x4+tg⁡2xdx\int \frac{\operatorname {ctg}^{3} x+\operatorname {ctg} x}{4+\operatorname {tg}^{2} x} d x.

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Problem 2.5.118

Find the integral. ∫dx(1+cos⁡2x)2\int \frac{d x}{\left(1+\cos^{2} x\right)^{2}}.

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Problem 2.5.119

Find the integral. ∫(sin⁡2xsin⁡3x+cos⁡3x)2dx\int \left(\frac{\sin 2 x}{\sin^{3} x+\cos^{3} x}\right)^{2} d x.

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Problem 2.5.120

Find the integral. ∫dxsin⁡x+2cos⁡x+3\int \frac{d x}{\sin x+2 \cos x+3}.

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Problem 2.5.121

Find the integral. ∫dx3−4sin⁡2x+2cos⁡2x\int \frac{d x}{3-4 \sin 2 x+2 \cos^{2} x}.

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Problem 2.5.122

Find the integral. ∫1−sin⁡2xdx\int \sqrt{1-\sin 2 x} d x.

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Problem 2.5.123

Find the integral. ∫dx1+cos⁡x,x∈(0;π)\int \frac{d x}{\sqrt{1+\cos x}}, x \in (0 ; \pi ).

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Problem 2.5.124

Find the integral. ∫th⁡xdx1−th⁡x\int \frac{\operatorname {th} x d x}{\sqrt{1-\operatorname {th} x}}.

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Problem 2.5.125

Find the integral. ∫sh⁡xdxch⁡2x\int \frac{\operatorname {sh} x d x}{\sqrt{\operatorname {ch} 2 x}}

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Problem 2.5.126

Find the integral. ∫sin⁡32xsin⁡5xdx\int \frac{\sqrt{\sin^{3} 2 x}}{\sin^{5} x} d x.

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Problem 2.5.127

Find the integral. ∫tg⁡xdx1+sin⁡2x\int \frac{\operatorname {tg} x d x}{\sqrt{1+\sin^{2} x}}.

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Problem 2.5.128

Find the integral. ∫sin⁡xdx2+sin⁡2x\int \frac{\sin x d x}{\sqrt{2+\sin 2 x}}.

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Problem 2.5.129

Find the integral. ∫x3sin⁡x2dx\int x^{3} \sin x^{2} d x.

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Problem 2.5.130

Find the integral. ∫xctg⁡2xdx\int x \operatorname {ctg}^{2} x d x.

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Problem 2.5.131

Find the integral. ∫xdx1+cos⁡x\int \frac{x d x}{1+\cos x}.

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Problem 2.5.132

Find the integral. ∫xsin⁡xdx(1+cos⁡x)2\int \frac{x \sin x d x}{(1+\cos x)^{2}}.

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Problem 2.5.133

Find the integral. ∫xdxsin⁡4x\int \frac{x d x}{\sin^{4} x}.

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Problem 2.5.134

Find the integral. ∫xsin⁡xdx(4−sin⁡2x)3\int \frac{x \sin x d x}{\sqrt{\left(4-\sin^{2} x\right)^{3}}}.

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Problem 2.5.135

Find the integral. ∫e2xdx1+ex\int \frac{e^{2 x} d x}{1+e^{x}}.

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Problem 2.5.136

Find the integral. ∫1+e2x(1+ex)2dx\int \frac{1+e^{2 x}}{\left(1+e^{x}\right)^{2}} d x.

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Problem 2.5.137

Find the integral. ∫x+12xdx\int \frac{x+1}{2^{x}} d x.

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Problem 2.5.138

Find the integral. ∫xexdx(1+ex)2\int \frac{x e^{x} d x}{\left(1+e^{x}\right)^{2}}.

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Problem 2.5.139

Find the integral. ∫xexdx(x+1)2\int \frac{x e^{x} d x}{(x+1)^{2}}.

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Problem 2.5.140

Find the integral. ∫xexdx\int x e^{\sqrt{x}} d x.

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Problem 2.5.141

Find the integral. ∫e2xex+e2xdx\int e^{2 x} \sqrt{e^{x}+e^{2 x}} d x.

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Problem 2.5.142

Find the integral. ∫ex+exdx\int e^{x+e^{x}} d x.

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Problem 2.5.143

Find the integral. ∫1+sin⁡x1+cos⁡xexdx\int \frac{1+\sin x}{1+\cos x} e^{x} d x.

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Problem 2.5.144

Find the integral. ∫(sin⁡xex)2dx\int \left(\frac{\sin x}{e^{x}}\right)^{2} d x.

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Problem 2.5.145

Find the integral. ∫dxx1−ln⁡2x\int \frac{d x}{x \sqrt{1-\ln^{2} x}}.

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Problem 2.5.146

Find the integral. ∫(xln⁡x)2dx\int (x \ln x)^{2} d x.

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Problem 2.5.147

Find the integral. ∫(ln⁡xx)2dx\int \left(\frac{\ln x}{x}\right)^{2} d x.

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Problem 2.5.148

Find the integral. ∫x3ln⁡∣x+3x−3∣dx\int x^{3} \ln \left|\frac{x+3}{x-3}\right| d x.

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Problem 2.5.149

Find the integral. ∫1x2ln⁡∣x+2x−2∣dx\int \frac{1}{x^{2}} \ln \left|\frac{x+2}{x-2}\right| d x.

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Problem 2.5.150

Find the integral. ∫ln⁡∣x∣(x+2)2dx\int \frac{\ln \left|x\right|}{(x+2)^{2}} d x.

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Problem 2.5.151

Find the integral. ∫xln⁡∣x∣(1+x2)2dx\int \frac{x \ln \left|x\right|}{\left(1+x^{2}\right)^{2}} d x.

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Problem 2.5.152

Find the integral. ∫ln⁡(1+x2)x2dx\int \frac{\ln \left(1+x^{2}\right)}{x^{2}} d x

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Problem 2.5.153

Find the integral. ∫xln⁡(1−x+x2)(1+x2)2dx\int \frac{x \ln \left(1-x+x^{2}\right)}{\left(1+x^{2}\right)^{2}} d x.

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Problem 2.5.154

Find the integral. ∫ln⁡xx−1dx\int \frac{\ln x}{\sqrt{x-1}} d x.

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Problem 2.5.155

Find the integral. ∫ln⁡(1−x2)1−xdx\int \frac{\ln \left(1-x^{2}\right)}{\sqrt{1-x}} d x.

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Problem 2.5.156

Find the integral. ∫xln⁡∣x∣dx(1−x2)x2−1\int \frac{x \ln \left|x\right| d x}{\left(1-x^{2}\right) \sqrt{x^{2}-1}}.

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Problem 2.5.157

Find the integral. ∫ln⁡∣x+x2−1∣x2dx\int \frac{\ln \left|x+\sqrt{x^{2}-1}\right|}{x^{2}} d x.

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Problem 2.5.158

Find the integral. ∫xln⁡(x+x2+1)x2+1 dx\int \frac{x \ln \left(x+\sqrt{x^{2}+1}\right)}{\sqrt{x^{2}+1}} \, d x.

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Problem 2.5.159

Find the integral. ∫ln⁡(x+x2+1)(x2+1)dx\int \frac{\ln \left(x+\sqrt{x^{2}+1}\right)}{\left(x^{2}+1\right)} d x.

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Problem 2.5.160

Find the integral. ∫x1−x2ln⁡x1−xdx\int \frac{x}{\sqrt{1-x^{2}}} \ln \frac{x}{\sqrt{1-x}} d x.

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Problem 2.5.161

Find the integral. ∫xx(1+ln⁡x)dx\int x^{x}(1+\ln x) d x.

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Problem 2.5.162

Find the integral. ∫ln⁡xcos⁡ln⁡xxdx\int \frac{\ln x \cos \ln x}{x} d x.

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Problem 2.5.163

Find the integral. ∫sin⁡x(cos⁡x+2−sin⁡2x)dx\int \sin x\left(\cos x+\sqrt{2-\sin^{2} x}\right) d x.

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Problem 2.5.164

Find the integral. ∫arctg⁡x51+x2dx\int \frac{\sqrt[5]{\operatorname {arctg} x}}{1+x^{2}} d x.

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Problem 2.5.165

Find the integral. ∫(x2+x)arcctg⁡xdx\int \left(x^{2}+x\right) \operatorname {arcctg} x d x.

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Problem 2.5.166

Find the integral. ∫arctg⁡2xx2 dx\int \frac{\operatorname {arctg} 2 x}{x^{2}} \, d x.

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Problem 2.5.167

Find the integral. ∫2x2+1x2+1arctg⁡xdx\int \frac{2 x^{2}+1}{x^{2}+1} \operatorname {arctg} x d x.

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Problem 2.5.168

Find the integral. ∫x4arctg⁡xx2+1dx\int \frac{x^{4} \operatorname {arctg} x}{x^{2}+1} d x.

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Problem 2.5.169

Find the integral. ∫xarcctg⁡x(x2+1)2dx\int \frac{x \operatorname {arcctg} x}{\left(x^{2}+1\right)^{2}} d x.

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Problem 2.5.170

Find the integral. ∫xarcsin⁡(x−1)dx\int x \arcsin (x-1) d x.

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Problem 2.5.171

Find the integral. ∫x2arccos⁡2xdx\int x^{2} \arccos 2 x d x.

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Problem 2.5.172

Find the integral. ∫arccos⁡xx2dx\int \frac{\arccos x}{x^{2}} d x.

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Problem 2.5.173

Find the integral. ∫arcsin⁡3x3dx\int \arcsin^{3} \frac{x}{3} d x.

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Problem 2.5.174

Find the integral. ∫x3arccos⁡1xdx\int x^{3} \arccos \frac{1}{x} d x.

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Problem 2.5.175

Find the integral. ∫3x2−1xxarctg⁡xdx\int \frac{3 x^{2}-1}{x \sqrt{x}} \operatorname {arctg} x d x.

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Problem 2.5.176

Find the integral. ∫arctg⁡(1−x)dx\int \operatorname {arctg}(1-\sqrt{x}) d x.

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Problem 2.5.177

Find the integral. ∫arcctg⁡1−xdx\int \operatorname {arcctg} \sqrt{1-x} d x.

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Problem 2.5.178

Find the integral. ∫xarctg⁡xx+1dx\int \frac{\sqrt{x} \operatorname {arctg} \sqrt{x}}{x+1} d x.

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Problem 2.5.179

Find the integral. ∫1−x2arcsin⁡xdx\int \sqrt{1-x^{2}} \arcsin x d x.

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Problem 2.5.180

Find the integral. ∫arcsin⁡xdx(1−x2)1−x2\int \frac{\arcsin x d x}{\left(1-x^{2}\right) \sqrt{1-x^{2}}}.

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Problem 2.5.181

Find the integral. ∫xarcsin⁡xdx(x2−1)1−x2\int \frac{x \arcsin x d x}{\left(x^{2}-1\right) \sqrt{1-x^{2}}}.

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Problem 2.5.182

Find the integral. ∫(1+x2)arcsin⁡xx21−x2dx\int \frac{\left(1+x^{2}\right) \arcsin x}{x^{2} \sqrt{1-x^{2}}} d x.

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Problem 2.5.183

Find the integral. ∫xarcsin⁡1−xdx\int \sqrt{x} \arcsin \sqrt{1-x} d x.

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Problem 2.5.184

Find the integral. ∫arcsin⁡x1−xdx\int \frac{\arcsin \sqrt{x}}{\sqrt{1-x}} d x.

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Problem 2.5.185

Find the integral. ∫arcsin⁡x(x−1)2dx\int \frac{\arcsin \sqrt{x}}{(x-1)^{2}} d x.

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Problem 2.5.186

Find the integral. ∫x3arccos⁡x1−x2dx\int \frac{x^{3} \arccos x}{\sqrt{1-x^{2}}} d x.

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Problem 2.5.187

Find the integral. ∫arccos⁡xx+1dx\int \arccos \sqrt{\frac{x}{x+1}} d x.

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Problem 2.5.188

Find the integral. ∫x2arccos⁡(xx)(1−x3)2dx\int \frac{x^{2} \arccos (x \sqrt{x})}{\left(1-x^{3}\right)^{2}} d x.

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Problem 2.5.189

Find the integral. ∫cos⁡xarctg⁡sin⁡xdx\int \cos x \operatorname {arctg} \sin x d x.

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Problem 2.5.190

Find the integral. ∫arcsin⁡tg⁡xcos⁡2x dx\int \frac{\arcsin \operatorname {tg} x}{\cos^{2} x} \, d x.

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Problem 2.5.191

Find the integral. ∫exarcsin⁡exdx\int e^{x} \arcsin e^{x} d x.

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Problem 2.5.192

Find the integral. ∫sh⁡xarcsin⁡exdx\int \operatorname {sh} x \arcsin e^{x} d x.

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Problem 2.5.193

Find the integral. ∫sh⁡xarctg⁡sh⁡xdx\int \operatorname {sh} x \operatorname {arctg} \operatorname {sh} x d x.

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Problem 2.5.194

Find the integral. ∫earctg⁡x(1+x2)1+x2dx\int \frac{e^{\operatorname {arctg} x}}{\left(1+x^{2}\right) \sqrt{1+x^{2}}} d x.

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Problem 2.5.195

Find the integral. ∫arctg⁡ex(1+ex)exdx\int \frac{\operatorname {arctg} \sqrt{e^{x}}}{\left(1+e^{x}\right) \sqrt{e^{x}}} d x.

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Problem 2.5.196

Find the integral. ∫xarctg⁡xln⁡(1+x2)dx\int x \operatorname {arctg} x \ln \left(1+x^{2}\right) d x.

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Problem 2.5.197

Find the integral, depending on parameters. ∫dxax2+bx+c,a≠0\int \frac{d x}{a x^{2}+b x+c}, a \neq 0.

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Problem 2.5.198

Find the integral, depending on parameters. ∫dxx3+bx2+ax+ab,ab≠0\int \frac{d x}{x^{3}+b x^{2}+a x+a b}, \quad a b \neq 0.

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Problem 2.5.199

Find the integral, depending on parameters. ∫dx(x2+(a+b)x+ab)2,a≠b\int \frac{d x}{\left(x^{2}+(a+b) x+a b\right)^{2}}, a \neq b.

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Problem 2.5.200

Find the integral, depending on parameters. ∫dxx4+(a2+b2)x2+a2b2,ab≠0\int \frac{d x}{x^{4}+\left(a^{2}+b^{2}\right) x^{2}+a^{2} b^{2}}, a b \neq 0.

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Problem 2.5.201

Find the integral, depending on parameters. ∫(x+ax+b)ndx,n∈N\int \left(\frac{x+a}{x+b}\right)^{n} d x, n \in \mathbb {N}

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Problem 2.5.202

Find the integral, depending on parameters. ∫dxx2n−a2n,n∈N,a>0\int \frac{d x}{x^{2 n}-a^{2 n}}, n \in \mathbb {N}, a > 0.

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Problem 2.5.203

Find the integral, depending on parameters. ∫ax+b+cax+b+ddx,a≠0\int \frac{\sqrt{a x+b}+c}{\sqrt{a x+b+d}} d x, a \neq 0

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Problem 2.5.204

Find the integral, depending on parameters. ∫dxax2+b,a≠0\int \frac{d x}{\sqrt{a x^{2}+b}}, a \neq 0.

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Problem 2.5.205

Find the integral, depending on parameters. ∫dx−x2+(a+b)x−ab,a≠b\int \frac{d x}{\sqrt{-x^{2}+(a+b) x-a b}}, a \neq b.

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Problem 2.5.206

Find the integral, depending on parameters. ∫x2+(a+b)x−abdx,a<b,x>b\int \sqrt{x^{2}+(a+b) x-a b} d x, a < b, x > b.

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Problem 2.5.207

Find the integral, depending on parameters. ∫dxxx2+a\int \frac{d x}{x \sqrt{x^{2}+a}}

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Problem 2.5.208

Find the integral, depending on parameters. ∫dx(x2−a2)b2−x2,ab≠0\int \frac{d x}{\left(x^{2}-a^{2}\right) \sqrt{b^{2}-x^{2}}}, a b \neq 0.

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Problem 2.5.209

Find the integral, depending on parameters. ∫dx(x2+a2)x2+b2,ab≠0\int \frac{d x}{\left(x^{2}+a^{2}\right) \sqrt{x^{2}+b^{2}}}, a b \neq 0

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Problem 2.5.210

Find the integral, depending on parameters. ∫dxxxn+a,a≠0,n∈N\int \frac{d x}{x \sqrt{x^{n}+a}}, a \neq 0, n \in \mathbb {N}.

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Problem 2.5.211

Find the integral, depending on parameters. ∫dx(x−a)n+1(x−b)n−1n,n∈N\int \frac{d x}{\sqrt[n]{(x-a)^{n+1}(x-b)^{n-1}}}, n \in \mathbb {N}.

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Problem 2.5.212

Find the integral, depending on parameters. ∫(cos⁡axcos⁡bx)2dx,ab≠0\int (\cos a x \cos b x)^{2} d x, \quad a b \neq 0.

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Problem 2.5.213

Find the integral, depending on parameters. ∫sin⁡(x+b)sin⁡(x+a) dx\int \frac{\sin (x+b)}{\sin (x+a)} \, d x

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Problem 2.5.214

Find the integral, depending on parameters. ∫dxsin⁡x−sin⁡a\int \frac{d x}{\sin x-\sin a}.

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Problem 2.5.215

Find the integral, depending on parameters. ∫dx1+2acos⁡x+a2\int \frac{d x}{1+2 a \cos x+a^{2}}.

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Problem 2.5.216

Find the integral, depending on parameters. ∫(a+cos⁡x)dx1+2acos⁡x+a2\int \frac{(a+\cos x) d x}{1+2 a \cos x+a^{2}}.

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Problem 2.5.217

Find the integral, depending on parameters. ∫tg⁡xtg⁡(x+a)dx\int \operatorname {tg} x \operatorname {tg}(x+a) d x.

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Problem 2.5.218

Find the integral, depending on parameters. ∫dxtg⁡2x+a\int \frac{d x}{\operatorname {tg}^{2} x+a}.

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Problem 2.5.219

Find the integral, depending on parameters. ∫dxa2cos⁡2x+b2sin⁡2x,ab≠0\int \frac{d x}{a^{2} \cos^{2} x+b^{2} \sin^{2} x}, a b \neq 0.

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Problem 2.5.220

Find the integral, depending on parameters. ∫dx(acos⁡x+bsin⁡x)2,a2+b2≠0\int \frac{d x}{(a \cos x+b \sin x)^{2}}, a^{2}+b^{2} \neq 0.

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Problem 2.5.221

Find the integral, depending on parameters. ∫xdx(acos⁡x+bsin⁡x)2\int \frac{x d x}{(a \cos x+b \sin x)^{2}}.

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Problem 2.5.222

Find the integral, depending on parameters. ∫dx(a+(ax+b)tg⁡x)2,a≠0\int \frac{d x}{(a+(a x+b) \operatorname {tg} x)^{2}}, a \neq 0.

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Problem 2.5.223

Find the integral, depending on parameters. ∫(x(ax−b)sin⁡x+(a+bx)cos⁡x)2dx,a2+b2≠0\int \left(\frac{x}{(a x-b) \sin x+(a+b x) \cos x}\right)^{2} d x, a^{2}+b^{2} \neq 0.

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Problem 2.5.224

Find the integral, depending on parameters. ∫dxaex+be−x,ab≠0\int \frac{d x}{a e^{x}+b e^{-x}}, a b \neq 0.

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Problem 2.5.225

Find the integral, depending on parameters. ∫dxa+bex,ab≠0\int \frac{d x}{\sqrt{a+b e^{x}}}, a b \neq 0.

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Problem 2.5.226

Find the integral, depending on parameters. ∫sh⁡axch⁡bxdx,a2+b2≠0\int \operatorname {sh} a x \operatorname {ch} b x d x, a^{2}+b^{2} \neq 0.

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Problem 2.5.227

Find the integral, depending on parameters. ∫dxa+bch⁡x+csh⁡x,a2+b2+c2≠0\int \frac{d x}{a+b \operatorname {ch} x+c \operatorname {sh} x}, a^{2}+b^{2}+c^{2} \neq 0.

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Problem 2.5.228

Find the integral, depending on parameters. ∫sh⁡axcos⁡bxdx,a2+b2≠0\int \operatorname {sh} a x \cos b x d x, a^{2}+b^{2} \neq 0.

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Problem 2.5.229

Find the integral, depending on parameters. ∫ln⁡∣x2+a∣dx\int \ln \left|x^{2}+a\right| d x.

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Problem 2.5.230

Find the integral, depending on parameters. ∫ln⁡∣x+x2+a∣x2dx\int \frac{\ln \left|x+\sqrt{x^{2}+a}\right|}{x^{2}} d x.

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Problem 2.5.231

Find the integral, depending on parameters. ∫ln⁡xx+adx\int \frac{\ln x}{\sqrt{x+a}} d x.

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Problem 2.5.232

Find the integral, depending on parameters. ∫xa−1ln⁡b−1x(aln⁡x+b)dx\int x^{a-1} \ln^{b-1} x(a \ln x+b) d x.

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Problem 2.5.233

Find the integral, depending on parameters. ∫xarcsin⁡x(1+ax2)2dx\int \frac{x \arcsin x}{\left(1+a x^{2}\right)^{2}} d x.

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Problem 2.5.234

Find the integral, depending on parameters. ∫arctg⁡xdx(ax2+b)ax2+b,ab≠0\int \frac{\operatorname {arctg} x d x}{\left(a x^{2}+b\right) \sqrt{a x^{2}+b}}, a b \neq 0.

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