11.61: A Brief Table of Integrals (by Trench)

$\int u^{\alpha} du=\frac{u^{\alpha +1}}{\alpha +1}+c,\quad \alpha\neq -1$

$\int \frac{du}{u}=\ln |u|+c$

$\int\cos u\:du=\sin u+c$

$\int\sin u\:du=-\cos u+c$

$\int\tan u\:du=-\ln |\cos u|+c$

$\int\cot u\:du=\ln |\sin u|+c$

$\int\sec ^{2}u\:du=\tan u+c$

$\int\csc ^{2}u\:du=-\cot u+c$

$\int\sec u\: du=\ln |\sec u+\tan u|+c$

$\int\cos ^{2}u\:du =\frac{u}{2}+\frac{1}{4}\sin 2u+c$

$\int\sin ^{2}u\: du=\frac{u}{2}-\frac{1}{4}\sin 2u+c$

$\int\frac{du}{1+u^{2}}du=\tan ^{-1}u+c$

$\int\frac{du}{\sqrt{1-u^{2}}}du=\sin ^{-1}u+c$

$\int\frac{1}{u^{2}-1}du=\frac{1}{2}\ln |\frac{u-1}{u+1}|+c$

$\int\cosh u\:du =\sinh u+c$

$\int\sinh u\:du =\cosh u+c$

$\int u\:dv =uv-\int v\:du$

$\int u\cos u\: du=u\sin u+\cos u+c$

$\int u\sin u\: du=-u\cos u+\sin u+c$

$\int ue^{u}du=ue^{u}-e^{u}+c$

$\int e^{\lambda u}\cos\omega u\:du =\frac{e^{\lambda u}(\lambda\cos\omega u+\omega\sin\omega u)}{\lambda ^{2}+\omega ^{2}}+c$

$\int e^{\lambda u}\sin\omega u\:du =\frac{e^{\lambda u}(\lambda\sin\omega u+\omega\cos\omega u)}{\lambda ^{2}+\omega ^{2}}+c$

$\int\ln |u|\:du=u\ln |u|-u+c$

$\int u\ln |u|\:du =\frac{u^{2}\ln |u|}{2}-\frac{u^{2}}{4}+c$

$\int\cos\omega _{1}u\cos\omega _{2}u\:du =\frac{\sin (\omega _{1}+\omega _{2})u}{2(\omega _{1}+\omega _{2})}+\frac{\sin (\omega _{1}-\omega _{2})u}{2(\omega _{1}-\omega _{2})}+c\quad (\omega _{1}\neq\pm\omega _{2})$

$\int\sin\omega _{1}u\sin\omega _{2}u\:du =-\frac{\sin (\omega _{1}+\omega _{2})u}{2(\omega _{1}+\omega _{2})}+\frac{\sin (\omega _{1}-\omega _{2})u}{2(\omega _{1}-\omega _{2})}+c\quad (\omega _{1}\neq\pm\omega _{2})$

$\int\sin\omega _{1}u\cos\omega _{2}u\:du =-\frac{\cos(\omega _{1}+\omega _{2})u}{2(\omega _{1}+\omega _{2})}-\frac{\cos (\omega _{1}-\omega _{2})u}{2(\omega _{1}-\omega _{2})}+c\quad (\omega _{1}\neq\pm\omega _{2})$