Following is a list of most widely used integration formulas. Some other formulas can be found in any standard calculus textbook, such as of James Stewart's CALCULUS Early Transcendentals
Basic integration formulas or Table of integrals.
General Forms
$$\int u^n \ dv = \frac{u^{n+1}}{n+1}+C, \quad n \ne -1$$
$$\int u \ du = uv -\int v \ du$$
$$\int \frac{du}{u} = \ln |u| +C$$
$$\int e^u \ du = e^u +C$$
$$\int a^u \ du = \frac{a^u}{\ln a} +C$$
Trigonometric Forms
$$\int \sin u \ du = -\cos u +C$$
$$\int \cos u \ du = \sin u +C$$
$$\int \sec^2 u \ du = \tan u +C$$
$$\int \csc^2 u \ du = -\cot u +C$$
$$\int \sec u \tan u \ du = \sec u +C$$
$$\int \csc u \cot u \ du = -\csc u +C$$
$$\int \tan u \ du = \ln |\sec u| +C$$
$$\int \cot u \ du = \ln |\sin u| +C$$
$$\int \sec u \ du = \ln |\sec u + \tan u| +C$$
$$\int \csc u \ du = \ln |\csc u - \cot u| +C$$
Other Forms
$$\int \frac{du}{\sqrt{a^2-u^2}}= \sin^{-1} \frac{u}{a} +C$$
$$\int \frac{du}{a^2+u^2}= \frac{1}{a}\tan^{-1} \frac{u}{a} +C$$
$$\int \frac{du}{u \sqrt{u^2-a^2}}= \frac{1}{a}\sec^{-1} \frac{u}{a} +C$$
$$\int \frac{du}{a^2-u^2}= \frac{1}{2a}\ln \left |\frac{u+a}{u-a} \right |+C$$
$$\int \frac{du}{u^2- a^2}= \frac{1}{2a}\ln \left |\frac{u-a}{u+a} \right |+C$$