$$\int (ax+b)^{n}dx=\frac{(ax+b)^{n+1} }{a(n+1)}+c$$ $$\int \frac{dx}{ax+b}=\frac{1}{a}\ln \left| ax+b\right|+c=\frac{1}{a}\ln C \left| ax+b\right|$$ $$\int \frac{xdx}{ax+b}=\frac{1}{a^{2} }\left| ax-b(ax+b)\right|+C$$ $$\int \frac{x^{2} dx}{ax+b}=\frac{x^{2} }{2a}-\frac{bx}{a^{2} }+\frac{b^{2} }{a^{2} }\ln \left| ax+b\right|+C$$ $$\int \frac{dx}{x(ax+b)}=-\frac{1}{b}\ln\left| \frac {ax+b}{x}\right|+C$$ $$\int \frac{dx}{x^{2}(ax+b)}=-\frac{1}{bx}+\frac{a}{b^{2} }\ln\left| \frac {ax+b}{x}\right|+C$$ $$\int \frac{xdx}{(ax+b)^{2} }=\frac{1}{a^{2}}(\ln \left| ax+b\right|+\frac{b}{ax+b})+C$$ $$\int \frac{x^{2}dx}{(ax+b)^{2} }=\frac{1}{a^{2}}[ax-2b\ln \left| ax+b\right|-\frac{b^{2}}{ax+b}]+C$$ $$\int \frac{dx}{x(ax+b)^{2} }=\frac{1}{b(ax+b)}-\frac{1}{b^{2}}\ln \left| \frac{ax+b}{x}\right|+C$$ $$\int \frac{dx}{a^{2}x^{2}-b^{2} }=\frac{1}{2ab}\ln \left| \frac{ax-b}{ax+b}\right|+C$$ $$\int \frac{dx}{a^{2}x^{2}-b^{2} }=\frac{1}{2ab}\arg\frac{a}{bx}+C$$ $$\int x\sqrt{ax+b}dx=\frac{2}{15a^{2}}(3ax-2b\sqrt{(ax+b)^{2}}+C$$ $$\int \frac{dx}{x\sqrt{ax+b}}\begin{cases} =\frac{1}{\sqrt{b}}\ln \frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}}+C, b>0 \\ =\frac{2}{\sqrt{-b}}\arg\tan \frac{\sqrt{ax+b}}{-b}+C, b<0 \end{cases}$$ $$\int \sin ^{m}x\cos ^{n}xdx\begin{cases} &=-\frac{\sin ^{m-1}x\cos ^{n+1}x}{m+n}+\frac{m-1}{m+n}\int \sin ^{m-1}x\cos ^{n}xdx , m\neq -n , \\ &= \frac{\sin ^{m+1}x\cos ^{n-1}x}{m+n}+\frac{n-1}{m+n}\int \sin ^{m}x\cos ^{n-1}xdx , m\neq -n , \end{cases}$$
