Computing Improper Integrals

$\int_{-\infty }^{\infty } \frac{x^{2} d x}{\left(x^{2}+1\right)\left(x^{2}+9\right)}$;

$\int_{-\infty }^{\infty } \frac{x^{2}-x+2}{x^{4}-10 x^{2}-9} d x$;

$\int_{-\infty }^{\infty } \frac{d x}{x^{2}-2 i x-2}$;

$\int_{-\infty }^{\infty } \frac{x^{2}+1}{x^{4}+1} d x$;

$\int_{-\infty }^{\infty } \frac{x^{4}+1}{x^{6}+1} d x$;

$\int_{-\infty }^{\infty } \frac{x^{2} d x}{x^{4}+6 x^{2}+25}$;

$\int_{-\infty }^{+\infty } \frac{d x}{\left(x^{2}+9\right)\left(x^{2}+16\right)}$;

$\int_{-\infty }^{+\infty } \frac{d x}{\left(x^{2}+1\right)^{3}}$.

$\int_{-\infty }^{+\infty } \frac{x+x^{3}}{1+x^{4}} \sin 3 x d x$;

$\int_{-\infty }^{+\infty } \frac{\cos 4 x}{1+x^{6}} d x$;

$\int_{-\infty }^{+\infty } \frac{(x+1) \sin 2 x}{x^{2}-4 x+8} d x$;

$\int_{-\infty }^{+\infty } \frac{(x-2) \cos x}{x^{2}-6 x+10} d x$;

$\int_{-\infty }^{+\infty } \frac{(x+3) \sin \frac{x}{2}}{x^{2}+4 x+20} d x$;

$\int_{-\infty }^{+\infty } \frac{(x-4) \cos \frac{2}{3} x}{x^{2}-6 x+90} d x$;

$\int_{-\infty }^{+\infty } \frac{(2-x) \cos (3 x-2)}{x^{2}-2 x+2} d x$;

$\int_{-\infty }^{+\infty } \frac{(x-3) \sin (x-1)}{x^{2}+4 x+5} d x$;

$\int_{-\infty }^{+\infty } \frac{(2 x+3) \sin (x+5)}{x^{2}+4 x+8} d x$;

$\int_{-\infty }^{+\infty } \frac{(x+7) \cos (x+2)}{x^{2}+2 x+5} d x$;

$\int_{-\infty }^{+\infty } \frac{\left(x^{3}+7 x\right) \sin 2 x}{x^{4}+5 x^{2}+4} d x$;

$\int_{-\infty }^{+\infty } \frac{4 x^{2} \cos x}{x^{4}+8 x^{2}+16} d x$;

$\int_{-\infty }^{+\infty } \frac{\cos (3-8 x)}{4 x^{2}-7 x+5} d x$;

$\int_{-\infty }^{+\infty } \frac{\sin (3-6 x)}{3 x^{2}-4 x+3} d x$;

$\int_{-\infty }^{+\infty } \frac{x \sin (2-x)}{x^{2}+2} d x$;

$\int_{-\infty }^{+\infty } \frac{x \cos (1-2 x)}{x^{2}+4} d x$;

$\int_{-\infty }^{+\infty } \frac{\cos (1-2 x)}{x-4+\frac{5}{x}} d x$;

$\int_{-\infty }^{+\infty } \frac{\sin (1-3 x)}{x-2+\frac{5}{x}} d x$;

$\int_{-\infty }^{+\infty } \frac{x^{3} \cos (1-2 x)}{\left(2 x^{2}+1\right)\left(x^{2}+4\right)} d x$;

$\int_{-\infty }^{+\infty } \frac{x^{3} \sin (2-x)}{\left(x^{2}+2\right)^{2}} d x$.

v. p. $\int_{-\infty }^{\infty } \frac{1-e^{i \alpha x}}{x^{2}} d x$, a) $\alpha > 0$; b) $\alpha < 0$;

$\int_{0}^{+\infty } \frac{1-\cos \alpha x}{x^{2}} d x, \alpha > 0$;

$\int_{0}^{+\infty }\left(\frac{\sin x}{x}\right)^{n} d x, n = 2,3,4$;

$\int_{0}^{+\infty } \frac{\cos 2 a x-\cos 2 b x}{x^{2}} d x, a > 0, b > 0$;

$\int_{0}^{+\infty } \frac{\sin a x}{x\left(x^{2}+b^{2}\right)} d x, a > 0, \mathrm{Re}\left(b\right) > 0$;

$\int_{0}^{+\infty } \frac{x^{2}-b^{2}}{x^{2}+b^{2}} \frac{\sin a x}{x} d x, a > 0, \mathrm{Re}\left(b\right) > 0$;

$\int_{0}^{+\infty } \frac{\sin^{2} a x}{x^{2}\left(x^{2}+b^{2}\right)} d x, a > 0, \mathrm{Re}\left(b\right) > 0$;

$\int_{0}^{+\infty } \frac{x-\sin x}{x^{3}\left(x^{2}+a^{2}\right)} d x$, Re $a > 0$.

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

$\int_{0}^{2} \frac{d x}{\sqrt[3]{x^{2}(2-x)}}$;

$\int_{0}^{1} \frac{d x}{(x-3)^{2} \sqrt{x-x^{2}}}$;

$\int_{0}^{6} \frac{1}{x+2} \sqrt{\frac{x}{6-x}} d x$;

$\int_{0}^{5} \frac{\sqrt[4]{x(5-x)^{7}}}{x^{2}-5 x-6} d x$;

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

$\int_{0}^{3} \frac{d x}{(x+2)^{2} \sqrt[4]{x^{3}(3-x)}}$;

$\int_{0}^{2} \frac{\sqrt[3]{x^{2}(2-x)}}{(x+2)^{2}} d x$;

$\int_{0}^{3} \sqrt[3]{(3-x) x^{2}} d x$;

$\int_{0}^{1} \sqrt{x-x^{2}} \frac{20 x+39}{x+2} d x$;

$\int_{-1}^{1} \sqrt[4]{\frac{1+x}{1-x}} \frac{3 x^{2}+13 x+20}{(x+2)^{2}} d x$;

$\int_{-1}^{0} \sqrt{-1-\frac{1}{x}} \frac{3 x+1}{3 x+4} d x$.

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

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

$\int_{0}^{+\infty } \frac{\sqrt{x} \ln x d x}{x^{2}+1}$;

$\int_{0}^{+\infty } \frac{\ln x}{(x+1) \sqrt[4]{x^{3}}} d x$;

$\int_{0}^{+\infty } \frac{\ln x d x}{(x+1)(2 x+1) \sqrt{x}}$;

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

$\int_{0}^{+\infty } \frac{d x}{\sqrt[5]{x}(x+1)^{2}}$;

$\int_{0}^{+\infty } \frac{\sqrt{x} \ln x}{(x+1)(x+2)} d x$.

$\int_{0}^{+\infty } \frac{\ln x}{(x+1)^{2}} d x$;

$\int_{0}^{+\infty } \frac{\ln x d x}{x^{2}+2 x+2}$;

$\int_{0}^{+\infty } \frac{\ln x d x}{\left(x^{2}+1\right)^{2}}$;

$\int_{0}^{+\infty } \frac{\ln x d x}{(x+1)\left(x^{2}+1\right)}$;

$\int_{0}^{+\infty } \frac{x \ln x}{\left(x^{2}+1\right)^{2}} d x$;

$\int_{0}^{+\infty } \frac{x^{2} \ln x}{\left(x^{2}+1\right)^{2}} d x$;

$\int_{0}^{+\infty } \frac{\ln x}{x^{2}-1} d x$;

$\int_{0}^{+\infty }\left(\frac{\ln x}{x-1}\right)^{2} d x$;

$\int_{0}^{+\infty } \frac{\ln^{2} x}{\left(x^{2}+1\right)^{2}} d x$;

$\int_{0}^{+\infty }\left(\frac{\ln x}{x-1}\right)^{3} d x$;

$\int_{-\infty }^{+\infty } \frac{x}{\operatorname {sh} x} d x$;

$\int_{0}^{+\infty } \frac{x d x}{\operatorname {sh} x \operatorname {ch} x}$;

$\int_{0}^{+\infty }\left(\frac{x}{\operatorname {sh} x}\right)^{2} d x$;

$\int_{0}^{+\infty }\left(\frac{x}{\operatorname {sh} x}\right)^{3} d x$;

$\int_{0}^{+\infty } \frac{x^{2}}{\operatorname {ch} x} d x$;

$\int_{0}^{1} \ln \frac{1-x}{x} \frac{d x}{1+x^{2}}$;

$\int_{0}^{1}\left(\ln \frac{x}{1-x}\right)^{2} \frac{d x}{x+1}$;

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

$\int_{0}^{+\infty } \frac{d x}{\left(x^{2}+\pi^{2}\right) \operatorname {ch} x}$;

$\int_{-\infty }^{+\infty } \frac{e^{x} d x}{\left(x^{2}+\pi^{2}\right) \operatorname {ch} x}$;

$\int_{0}^{+\infty } \frac{d x}{\left(x^{2}+9 \pi^{2}\right) \operatorname {ch} x}$;

$\int_{0}^{+\infty } \frac{d x}{(x+1)\left(\ln^{2} x+\pi^{2}\right)}$;

$\int_{-1}^{1} \frac{d x}{\left(\ln \frac{1+x}{1-x}\right)^{2}+9 \pi^{2}}$;

$\int_{0}^{1} \frac{x d x}{\left(\ln \frac{x}{1-x}\right)^{2}+\pi^{2}}$;

$\int_{0}^{+\infty } \frac{\ln x d x}{x^{2}+2 a x \cos \lambda +a^{2}}, a > 0, a < \lambda < \pi$.

$\int_{-\infty }^{+\infty } R(x) \ln \left|x-a\right| d x = -2 \pi \mathrm{Im}\left(\sum_{s = 1}^{n}\right) \mathop{\mathrm{res}}_{z = a_{s}} R(z) g(z)$ (here $a$ is a real number, and $g(z)$ is an arbitrary branch of the multivalued function $\operatorname {Ln}(z-a)$, regular in the upper half-plane);

$\int_{-\infty }^{+\infty } R(x)\left|x-a\right|^{\alpha -1} d x = -\frac{2 \pi }{\sin \frac{\pi \alpha }{2}} \mathrm{Im}\left(\sum_{s = 1}^{n}\right) \mathop{\mathrm{res}}_{z = a_{s}} R(z) g(z)$ (here $a$ is a real number, $\alpha > 0$, and $g(z)$ is the branch of the multivalued function $\left(\frac{z-a}{i}\right)^{\alpha -1}$, regular in the upper half-plane and positive for positive values of $\frac{z-a}{i}$);

$\int_{-\infty }^{+\infty } R(x) \ln \left|x^{2}-a^{2}\right| d x = -2 \pi \mathrm{Im}\left(\sum_{s = 1}^{n}\right) \mathop{\mathrm{res}}_{z = a_{s}} R(z) g(z)$ (here $a > 0$, and $g(z)$ is an arbitrary branch of the multivalued function $\operatorname {Ln}\left(z^{2}-a^{2}\right)$, regular in the upper half-plane);

$\int_{-\infty }^{+\infty } R(x) \ln \left(x^{2}+a^{2}\right) d x = -4 \pi \mathrm{Im}\left(\sum_{s = 1}^{n}\right) \mathop{\mathrm{res}}_{z = a_{s}} R(z) g(z)$ (here $a > 0$, and $g(z)$ is an arbitrary branch of the multivalued function $\operatorname {Ln}(z+a i)$, regular in the upper half-plane).

$\int_{-\infty }^{+\infty } \frac{\sqrt[3]{\left|x-1\right|}}{x^{2}+1} d x$;

$\int_{-\infty }^{+\infty } \frac{\ln \left|x^{2}-1\right|}{x^{2}+1} d x$;

$\int_{-\infty }^{+\infty } \frac{\ln \left(x^{4}+2 x^{2}+2\right)}{x^{2}+4} d x$;

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

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

$\int_{0}^{+\infty } \frac{\ln \left(x^{2}+2 x+2\right)}{\left(x^{2}+1\right)^{2}} d x$;

$\int_{0}^{+\infty } \frac{\ln \left(1+x^{2}\right)}{x^{2}} \frac{d x}{1+x^{2}}$;

$\int_{0}^{+\infty } \frac{\operatorname {arctg} x}{x} \frac{d x}{1+x^{2}}$;

$\int_{0}^{+\infty }(\operatorname {arctg} x)^{2} \frac{d x}{x^{2}}$;

$\int_{-\infty }^{+\infty } \frac{\operatorname {arctg} x}{x^{2}-4 x+5} \ln \left(x^{2}+1\right) d x$;

$\int_{0}^{+\infty } \frac{\sqrt{\sqrt{x^{2}+1}+x}-\sqrt{\sqrt{x^{2}+1}-x}}{x\left(x^{2}+1\right)} d x$;

$\int_{0}^{+\infty }\left(\sqrt{\sqrt{1+x^{2}}+x}+\sqrt{\sqrt{1+x^{2}}-x}-2\right) \frac{d x}{x^{2}}$;

$\int_{0}^{+\infty } \frac{\sqrt{\sqrt{1+x^{2}}+x}-\sqrt{\sqrt{1+x^{2}}-x}}{x \sqrt{1+x^{2}}} d x$;

$\int_{0}^{+\infty } \sqrt{\sqrt{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}+x^{2}-a b} \frac{d x}{\left(x^{2}+1\right)^{2}}, a > 0, b > 0$.

$\int_{0}^{+\infty } \frac{x d x}{\left(4 x^{2}+\pi^{2}\right) \operatorname {sh} x}$;

$\int_{-\infty }^{+\infty } \frac{e^{x} d x}{\left(4 x^{2}+\pi^{2}\right) \operatorname {ch} x}$;

$\int_{0}^{+\infty } \frac{\operatorname {sh}^{2} x d x}{\left(4 x^{2}+\pi^{2}\right) \operatorname {ch}^{2} x}$;

$\int_{0}^{+\infty } \frac{x d x}{\left(4 x^{2}+9 \pi^{2}\right) \operatorname {sh} x}$;

$\int_{0}^{+\infty } \frac{\ln x d x}{\left(x^{2}+1\right)\left(4 \ln^{2} x+9 \pi^{2}\right)}$;

$\int_{0}^{+\infty } \frac{\sqrt{x} \ln x}{\left(x^{2}-1\right)\left(\ln^{2} x+\pi^{2}\right)} d x$;

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

$\int_{1}^{+\infty } \ln \frac{x+\sqrt{x^{2}-1}}{x-\sqrt{x^{2}-1}} \frac{d x}{x^{2}+1}$;

$\int_{0}^{+\infty } \frac{\ln x d x}{\left(x^{2}+a^{2}\right)\left(4 \ln^{2} x+\pi^{2}\right)}, a > 0$;

$\int_{a}^{+\infty } \frac{\ln \left(x+\sqrt{x^{2}-a^{2}}\right)}{x^{2}+a^{2}} d x, a > 0$;

$\int_{a}^{+\infty } \operatorname {arctg} \frac{a}{x} \frac{d x}{\sqrt{x^{2}-a^{2}}}, a > 0$;

$\int_{0}^{+\infty } \frac{\ln \left(e^{x}+\sqrt{e^{2 x}-e^{2 a}}\right)}{\operatorname {ch} x} d x$.