
# A Brief Table of Integrals


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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