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Integration formulas

rjahan's picture
by rjahan | February 5, 2010 - 7:00am

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

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

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

$$\int \frac{du}{u \sqrt{u^2-a^2}}= \frac{1}{a}\sec^{-1} \frac{u}{a} +C$$

$$\int \frac{du}{a^2-u^2}= \frac{1}{2a}\ln \left |\frac{u+a}{u-a} \right |+C$$

$$\int \frac{du}{u^2- a^2}= \frac{1}{2a}\ln \left |\frac{u-a}{u+a} \right |+C$$