Various examples of the integration of functions

Evaluate the integral: $\displaystyle \int \dfrac {dx}{x^{6}(1+x^{2})}$.

Evaluate the integral: $\displaystyle \int \dfrac {x^{2}\, dx}{(1-x^{2})^{3}}$.

Evaluate the integral: $\displaystyle \int \dfrac {dx}{1+x^{4}+x^{8}}$.

Evaluate the integral: $\displaystyle \int \dfrac {dx}{\sqrt{x}+\sqrt[3]{x}}$.

Evaluate the integral: $\displaystyle \int x^{2}\sqrt{\dfrac {x}{1-x}}\, dx$.

Evaluate the integral: $\displaystyle \int \dfrac {x+2}{x^{2}\sqrt{1-x^{2}}}\, dx$.

Find the integral: $\displaystyle \int \sqrt{\dfrac {x}{1-x\sqrt{x}}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {x^{5}\, dx}{\sqrt{1+x^{2}}}$.

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt[3]{x^{2}(1-x)}}$.

Find the integral: $\displaystyle \int \dfrac {dx}{x\sqrt{1+x^{3}+x^{6}}}$.

Find the integral: $\displaystyle \int \dfrac {dx}{x\sqrt{x^{4}-2x^{2}-1}}$.

Find the integral: $\displaystyle \int \dfrac {1+\sqrt{1-x^{2}}}{1-\sqrt{1-x^{2}}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {(1+x)\, dx}{x+\sqrt{x+x^{2}}}$.

Find the integral: $\displaystyle \int \dfrac {\ln (1+x+x^{2})}{(1+x)^{2}}\, dx$.

Find the integral: $\displaystyle \int (2x+3)\arccos (2x-3)\, dx$.

Find the integral: $\displaystyle \int x\ln (4+x^{4})\, dx$.

Find the integral: $\displaystyle \int \dfrac {\arcsin x}{x^{2}}\cdot \dfrac {1+x^{2}}{\sqrt{1-x^{2}}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {x\ln (1+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}}\, dx$.

Find the integral: $\displaystyle \int x\sqrt{x^{2}+1}\ln \sqrt{x^{2}-1}\, dx$.

Find the integral: $\displaystyle \int \dfrac {x}{\sqrt{1-x^{2}}}\ln \dfrac {x}{\sqrt{1-x}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {dx}{(2+\sin x)^{2}}$.

Find the integral: $\displaystyle \int \dfrac {\sin 4x}{\sin^{8}x+\cos^{8}x}\, dx$.

Find the integral: $\displaystyle \int \dfrac {dx}{\sin x\sqrt{1+\cos x}}$.

Find the integral: $\displaystyle \int \dfrac {ax^{2}+b}{x^{2}+1}\operatorname {arctg}x\, dx$.

Find the integral: $\displaystyle \int \dfrac {ax^{2}+b}{x^{2}-1}\ln \left|\dfrac {x-1}{x+1}\right|\, dx$.

Find the integral: $\displaystyle \int \dfrac {x\ln x}{(1+x^{2})^{2}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {x\operatorname {arctg}x}{\sqrt{1+x^{2}}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {\sin 2x\, dx}{\sqrt{1+\cos^{4}x}}$.

Find the integral: $\displaystyle \int \dfrac {x^{3}\arccos x}{\sqrt{1-x^{2}}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {x^{4}\operatorname {arctg}x}{1+x^{2}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {x\operatorname {arcctg}x}{(1+x^{2})^{2}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {x\ln (x+\sqrt{1+x^{2}})}{(1-x^{2})^{2}}\, dx$.

Find the integral: $\displaystyle \int \sqrt{1-x^{2}}\arcsin x\, dx$.

Find the integral: $\displaystyle \int x(1+x^{2})\operatorname {arcctg}x\, dx$.

Find the integral: $\displaystyle \int x^{x}(1+\ln x)\, dx$.

Find the integral: $\displaystyle \int \dfrac {\arcsin e^{x}}{e^{x}}\, dx$.

Find the integral: $\displaystyle \int \dfrac {\operatorname {arctg}e^{\frac{x}{2}}}{e^{\frac{x}{2}}(1+e^{x})}\, dx$.

Find the integral: $\displaystyle \int \dfrac {dx}{(e^{x+1}+1)^{2}-(e^{x-1}+1)^{2}}$.

Find the integral: $\displaystyle \int \sqrt{\operatorname {th}^{2}x+1}\, dx$.

Find the integral: $\displaystyle \int \dfrac {1+\sin x}{1+\cos x}\cdot e^{x}\, dx$.

Find the integral: $\displaystyle \int \left|x\right|\, dx$.

Find the integral: $\displaystyle \int x\left|x\right|\, dx$.

Find the integral: $\displaystyle \int (x+\left|x\right|)^{2}\, dx$.

Find the integral: $\displaystyle \int (\left|1+x\right|-\left|1-x\right|)\, dx$.

Find the integral: $\displaystyle \int e^{-\left|x\right|}\, dx$.

Find the integral: $\displaystyle \int \max (1,x^{2})\, dx$.

Find the integral: $\displaystyle \int \varphi (x)\, dx$, where $\varphi (x)$ is the distance from the number $x$ to its nearest integer.

Find the integral: $\displaystyle \int [x]\left|\sin \pi x\right|\, dx \quad (x\geqslant 0)$.

Find the integral: $\displaystyle \int f(x)\, dx$, where

Find the integral: $\displaystyle \int f(x)\, dx$, where

Find the integral: $\displaystyle \int xf''(x)\, dx$.

Find the integral: $\displaystyle \int f'(2x)\, dx$.

Suppose $f'(x^{2})=\dfrac {1}{x}$ ($x>0$), find $f(x)$.

Suppose $f'(\sin^{2}x)=\cos^{2}x$, find $f(x)$.

Suppose

and $f(0)=0$, find $f(x)$.