Math Problems Vault

Problem 2.3.1

Find the integral.

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(1)

$\int \frac{\sqrt{x} d x}{1+\sqrt{x}}$ ;

(2)

$\int \frac{1-2 \sqrt{x}}{1+2 \sqrt{x}} d x$

(3)

$\int \frac{\sqrt{x+1}+1}{\sqrt{x+1}-1} d x$ ;

(4)

$\int \frac{\sqrt{x-1}-\sqrt{x+1}}{\sqrt{x-1}+\sqrt{x+1}} d x$ ;

(5)

$\int \sqrt[3]{\frac{x+1}{x-1}} d x$

(6)

$\int \frac{x d x}{\sqrt[4]{x^{3}(4-x)}}$ .

Problem 2.3.2

Find the integral.

?

(1)

$\int \frac{\sqrt{x+4}}{x} d x$ ;

(2)

$\int \frac{d x}{3 x+\sqrt[3]{x^{2}}}$ ;

(3)

$\int x \sqrt[4]{x-2} d x$ ;

(4)

$\int \frac{x \sqrt[3]{x+2}}{x+\sqrt[3]{x+2}} d x$ ;

(5)

$\int x \sqrt{\frac{x-1}{x+1}} d x$ ;

(6)

$\int \sqrt[5]{\frac{x}{x+1}} \frac{d x}{x^{3}}$ ;

(7)

$\int \frac{d x}{\sqrt[6]{(x-7)^{7}(x-5)^{5}}}$ ;

(8)

$\int \sqrt[n]{\sqrt{(x-a)^{n+1}(x-b)^{n-1}}}, a \neq b$ .

Problem 2.3.3

Prove that the integral $\int R\left(x ; \sqrt[n]{(x-a)^{p}(x-b)^{q}}\right) d x$ , where $R$ is a rational function of the variables $x$ and $y = \sqrt[n]{(x-a)^{p}(x-b)^{q}}, p, q \in \mathbb {Z}, \quad a, b \in \mathbb {R}$ , under the condition $(p+q) / n \in \mathbb {Z}$ , is an elementary function.

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Problem 2.3.4

Find the integral.

?

(1)

$\int \frac{d x}{\sqrt{x}+\sqrt[3]{x^{2}}}$ ;

(2)

$\int -\frac{\sqrt[6]{x} d x}{1+\sqrt[3]{x}}$

(3)

$\int \frac{d x}{\sqrt[3]{4 x^{2}+4 x+1}-\sqrt{2 x+1}}$ ;

(4)

$\int \frac{d x}{\left(\sqrt[6]{x}+\sqrt[12]{x^{5}}\right)^{3}}$ ;

(5)

$\int \frac{d x}{x+2 \sqrt{x^{3}}+\sqrt[3]{x^{4}}}$ ;

(6)

$\int \frac{d x}{2 \sqrt{x}-\sqrt[3]{x-\sqrt[4]{x}}}$ .

Problem 2.3.5

Find the integral.

?

(1)

$\int -\frac{1-x+x^{2}}{\sqrt{1+x-x^{2}}} d x$ ;

(2)

$\int \frac{2 x^{2}-3 x}{\sqrt{x^{2}-2 x+5}} d x$ ;

(3)

$\int \frac{x^{3} d x}{\sqrt{x^{2}+x+1}}$ ;

(4)

$\int \frac{x^{3}+2 x^{2}+x-1}{\sqrt{x^{2}+2 x-1}} \cdot d x$ ;

(5)

$\int \frac{x^{3}-6 x^{2}+11 x-6}{\sqrt{x^{2}+4 x+3}} d x$

(6)

$\int \sqrt{3-4 x+4 x^{2}} d x$ ;

(7)

$\int x \sqrt{x^{2}+2 x+2} d x$

(8)

$\int x^{2} \sqrt{x^{2}+4} d x$

Problem 2.3.6

Under what condition is the integral $\int \frac{a_{1} x^{2}+b_{1} x+c_{1}}{\sqrt{a x^{2}+b x+c}} d x, a \neq 0$ , an algebraic function?

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Problem 2.3.7

For the integral $J_{n} = \int \frac{x^{n} d x}{\sqrt{a x^{2}+b x+c}}, n \in \mathbb {N}, \quad n > 1$ , prove the recurrence formula

$J_{n} = \frac{1}{n a}\left(x^{n-1} \sqrt{a x^{2}+b x+c}-\frac{b}{2}(2 n-1) J_{n-1}-c(n-1) J_{\ddot{n}-2}\right)$ .

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Problem 2.3.8

(1)(2)(3)(4)

Find the integral.

?

(1)

$\int \frac{x^{3} d x}{\sqrt{1+2 x-x^{2}}}$ ;

(2)

$\int \frac{x^{4} d x}{\sqrt{x^{2}+4 x+5}}$ ;

(3)

$\int \frac{x^{6} d x}{\sqrt{x^{2}+1}}$ ;

(4)

$\int \frac{x^{8} d x}{\sqrt{x^{2}-1}}$ .

Problem 2.3.9

(1)(2)(3)(4)

Find the integral.

?

(1)

$\int \frac{d x}{x \sqrt{5 x^{2}-2 x+1}}, x > 0$ ;

(2)

$\int \frac{d x}{(x+1) \sqrt{x^{2}+x+1}}, x > -1$ ;

(3)

$\int \frac{d x}{x^{3} \sqrt{1+2 x+2 x^{2}}}, \quad x > 0$ ;

(4)

$\int \frac{d x}{(x-1)^{3} \sqrt{x^{2}-2 x-1}}, x > 1+\sqrt{2}$ .

Problem 2.3.10

(1)(2)(3)(4)

Find the integral.

?

(1)

$\int \frac{d x}{\left(x^{2}+4 x+7\right)^{3 / 2}}$ ;

(2)

$\int \frac{d x}{\left(x^{2}+x+1\right)^{5 / 2}} ;$

(3)

$\int \frac{d x}{\left(x^{2}+1\right)^{7 / 2}}$ ,

(4)

$\int \frac{(x+1) d x}{\left(x^{2}+x+1\right)^{3 / 2}}$ .

Problem 2.3.11

(1)(2)

Find the integral.

?

(1)

$\int \frac{d x}{\left(x^{2}+1\right) \sqrt{3 x^{2}+1}}$ ;

(2)

$\int \frac{x d x}{\left(2 x^{2}+1\right) \sqrt{3 x^{2}+5}}$ .

Problem 2.3.12

(1)(2)(3)(4)

Find the integral.

?

(1)

$\int \frac{(2 x+3) d x}{\left(x^{2}+2 x+3\right) \sqrt{x^{2}+2 x+4}}$ ;

(2)

$\int \frac{(x+3) \, d x}{\left(x^{2}+1\right) \sqrt{x^{2}+x+1}}$ ;

(3)

$\int \frac{d x}{\left(x^{2}-x+1\right) \sqrt{x^{2}+x+1}}$ ;

(4)

$\int \frac{x d x}{\left(3 x^{2}+2 x+3\right) \sqrt{2 x^{2}-x+2}}$ .

Problem 2.3.13

Find the integral.

?

(1)

$\int \frac{(3 x+2) d x}{(x+1) \sqrt{x^{2}+3 x+3}}, x > -1$ ;

(2)

$\int \frac{x^{2}+x+1}{x \sqrt{x^{2}-x+1}} d x, x > 0$ ;

(3)

$\int \frac{d x}{\left(x^{2}+x-2\right) \sqrt{x^{2}+2 x+3}}, x > 1$ ;

(4)

$\int \frac{d x}{\left(x^{3}-x\right) \sqrt{x^{2}+x+4}}, \quad x > 1$ ;

(5)

$\int \frac{\sqrt{x^{2}+x+1}}{(x+1)^{2}} d x, \quad x > -1$ ;

(6)

$\int \frac{x^{3} d x}{(1+x) \sqrt{1+2 x-x^{2}}}$ ;

(7)

$\int \frac{\sqrt{x^{2}+1}}{x^{2}+2} d x$ ;

(8)

$\int \frac{x^{2} d x}{\left(4-2 x+x^{2}\right) \sqrt{2+2 x-x^{2}}}$ ;

(9)

$\int \frac{x+\sqrt{1+x+x^{2}}}{1+x+\sqrt{1+x+x^{2}} } d x$ .

Problem 2.3.14

Find the integral.

?

(1)

$\int \frac{d x}{x-\sqrt{x^{2}-x+1}}$ ;

(2)

$\int \frac{d x}{1+\sqrt{x^{2}+2 x+2}}$ ;

(3)

$\int \frac{d x}{1+\sqrt{1-2 x-x^{2}}}$ ;

(4)

$\int \frac{d x}{\left(1+\sqrt{x^{2}+x}\right)^{2}}$ ;

(5)

$\int \left(\frac{\sqrt{1+x+x^{2}}-1}{x}\right)^{2} \frac{d x}{\sqrt{1+x+x^{2}}}$ ;

(6)

$\int \frac{\sqrt{x^{2}+3 x+2}-x}{\sqrt{x^{2}+3 x+2}+x} d x$ .

Problem 2.3.15

Prove that the computation of the integral

\int R(x ; \sqrt{a x+b} ; \sqrt{c x+d}) d x

where $R(x ; u ; v)$ is a rational function of the variables $x, u, v$ , reduces to the computation of the integral $\int R_{1}(t) d t$ , where $R_{1}(t)$ is a rational function.

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Problem 2.3.16

(1)(2)

Find the integral.

?

(1)

$\int \frac{d x}{1+\sqrt{x}+\sqrt{1+x}}$ ;

(2)

$\int \frac{d x}{(\sqrt{2 x-3}+2 x-3) \sqrt{4-2 x}}$ .

Problem 2.3.17

(1)(2)(3)(4)

Find the integral.

?

(1)

$\int x^{-1 / 3}\left(1-x^{1 / 6}\right)^{-1} d x$ ;

(2)

$\int x^{1 / 2}\left(1+x^{1 / 3}\right)^{-2} d x$ ;

(3)

$\int x^{-2 / 3}\left(1+x^{1 / 3}\right)^{-3} d x$ ;

(4)

$\int x^{-1 / 2}\left(1+x^{1 / 4}\right)^{-10} d x$ .

Problem 2.3.18

Find the integral.

?

(1)

$\int x^{2} \sqrt[3]{(x+1)^{2}} d x$ ;

(2)

$\int \sqrt[3]{1+\sqrt[4]{x}} d x$ ;

(3)

$\int \frac{\sqrt[3]{1+\sqrt[4]{x}}}{\sqrt{x}} d x$ ;

(4)

$\int \frac{\sqrt[3]{x} \, d x}{\sqrt{1+\sqrt[3]{x}}}$

(5)

$\int \frac{x d x}{\sqrt{1+\sqrt[3]{x^{2}}}}$ ;

(6)

$\int \frac{d x}{x \sqrt[6]{x^{6}+1}}$ .

Problem 2.3.19

(1)(2)(3)(4)

Find the integral.

?

(1)

$\int \frac{d x}{x^{2} \sqrt[3]{\left(2+x^{3}\right)^{5}}}$ ;

(2)

$\int \frac{d x}{x^{3} \sqrt[3]{2-x^{3}}}$ ;

(3)

$\int \frac{d x}{\sqrt[3]{1+x^{3}}}$ ,

(4)

$\int \sqrt[3]{x-x^{3}} d x$

Problem 2.3.20

For which rational values of the parameter $q$ is the integral $\int \sqrt{1+x^{q}} d x$ an elementary function?

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Problem 2.3.21

For the integral $J_{m, p} = \int x^{m}\left(a x^{n}+b\right)^{p} d x$ prove the formula

$a(m+1+n p) J_{m, p} = x^{m+1-n}\left(a x^{n}+b\right)^{p+1}-b(m+1-n) J_{m-n, p}$ and use it to find the integral $\int \frac{x^{9} d x}{\sqrt[3]{1+x^{3}}}$ .

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Problem 2.3.22

(1)(2)(3)(4)

Find the pseudo-elliptic integral:

?

(1)

$\int \frac{d x}{x^{7} \sqrt{x^{4}+1}}$

(2)

$\int \frac{\left(x^{2}-1\right) \, d x}{\left(x^{2}+1\right) \sqrt{x^{4}+1}}$

(3)

$\int \frac{(x-1) \, d x}{(x+1) \sqrt{x^{3}+x^{2}+x}}$ ;

(4)

$\int \frac{\left(x^{2}+1\right) d x}{x \sqrt{x^{4}+3 x^{3}-2 x^{2}-3 x+1}}, x > 1$ .

Problem 2.3.23

Find the substitution by means of which the integral $\int R\left(x ; \sqrt{P_{3}(x)}\right) d x$ can be transformed into the integral

\int R_{1}\left(t ; \sqrt{P_{4}(t)}\right) d t

$R(x ; y), R_{1}(t ; u)$ — rational functions, $P_{3}(x), P_{4}(x)$ — polynomials of the third and fourth degree.

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Problem 2.3.24

Prove that, using a substitution of the form $x = (\alpha t+\beta ) /(t+1)$ , the integral $\int R\left(x ; \sqrt{P_{4}(x)}\right) d x$ can be transformed into the integral

\int R_{1}\left(t ; \sqrt{A\left(1+\lambda _{1} t^{2}\right)\left(1+\lambda _{2} t^{2}\right)}\right) d t

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Problem 2.3.25

Prove that any integral of the form $\int R\left(x ; \sqrt{P_{4}(x)}\right) d x$ can be expressed in terms of elementary functions and the integral

\int \frac{R_{1}\left(t^{2}\right) d t}{\sqrt{A\left(1+\lambda _{1} t^{2}\right)\left(1+\lambda _{2} t^{2}\right)}}

where $R_{1}(u)$ is a rational function.

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Problem 2.3.26

Prove that the integral

\int \frac{R\left(x^{2}\right) d x}{\sqrt{A\left(1+\lambda _{1} x^{2}\right)\left(1+\lambda _{2} x^{2}\right)}}, \quad A = \pm 1, \quad \left|\lambda _{1}\right| > \left|\lambda _{2}\right| > 0

using one of the following substitutions:

$\sqrt{\left|\lambda_{1}\right|} x = t, \quad \sqrt{\left|\lambda_{1}\right|} x = \sqrt{1-t^{2}}, \quad \sqrt{\left|\lambda_{1}\right|} x = \frac{t}{\sqrt{1-t^{2}}}$ ,

\sqrt{\left|\lambda _{1}\right|} x = \frac{1}{\sqrt{1-t^{2}}}, \quad \sqrt{\left|\lambda _{2}\right|} x = \sqrt{1+\frac{\left|\lambda _{2}\right|-\left|\lambda _{1}\right|}{\left|\lambda _{1}\right|} t^{2}}

is reduced to the form $\int \frac{R_{1}\left(t^{2}\right) d t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2} t^{2}\right)}}, k \in (0 ; 1)$ .

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Problem 2.3.27

Express the elliptic integral in terms of elementary functions and the functions $F(\varphi ; k)$ and $E(\varphi ; k)$ (see (13), (14)):

?

(1)

$\int \frac{x^{2} d x}{\sqrt{36 x^{4}-13 x^{2}+1}}$

(2)

$\int \frac{d x}{\sqrt{1-2 x^{2}-8 x^{4}}}$ ;

(3)

$\int \frac{d x}{\sqrt{1+29 x^{2}+100 x^{4}}}$ ;

(4)

$\int \frac{d x}{x^{2} \sqrt{144 x^{4}+7 x^{2}-1}}$ ;

(5)

$\int \frac{d x}{\sqrt{-9 x^{4}+10 x^{2}-1}}$ ;

(6)

$\int \frac{d x}{\sqrt{1+x^{3}}}$ .