Subject: Calculus

List Of Integrals Of Trigonometric Functions

The following consists of the integrals of trigonometric functions and its combinations, products and variations.

The Page at a Glance

Integrals of the 6 trigonometric functions

  • \int \sin{u} du = -\cos{u} + C
  • \int \cos{u} du = \sin{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

Integrals of the variations of the 6 Trigo Functions

  • \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 \sin^{2}{u} du = \frac{1}{2}u - \frac{1}{4}\sin{2u}+C
  • \int \cos^{2}{u} du = \frac{1}{2}u + \frac{1}{4}\sin{2u}+C
  • \int \tan^{2}{u} du = -u + \tan{u} + C
  • \int \cot^{2}{u} du = -u \cot{u} + C

Integrals of trigonometric functions with n - exponents

  • \int \sin^{n}{u} du = -\frac{1}{n} \sin^{n-1}{u} \cos{u} + \frac{n-1}{n} \int \sin^{n-2}{u}du
  • \int \cos^{n}{u} du = \frac{1}{n} \cos^{n-1}{u} \sin{u} + \frac{n-1}{n} \int \cos^{n-2}{u}du
  • \int \tan^{n}{u} du = \frac{1}{n-1} \tan^{n-1}{u} - \int \tan^{n-2}{u}du
  • \int \cot^{n}{u} du = - \frac{1}{n-1} \cot^{n-1}{u} - \int \cot^{n-2}{u}du
  • \int \sec^{n}{u} du = \frac{1}{n-1} \sec^{n-2}{u} \tan{u} + \frac{n-2}{n-1} \int \sec^{n-2}{u}du
  • \int \csc^{n}{u} du = -\frac{1}{n-1} \csc^{n-2}{u} \cot{u} + \frac{n-2}{n-1} \int \csc^{n-2}{u}du

Integrals of trigonometric functions (products of sine and cosine)

  • \int \sin(mu)\sin(nu) du = -\frac{\sin{[m+n]u}}{2[m+n]} + \frac{\sin{[m-n]u}}{2[m-n]} + C
  • \int \cos(mu)\cos(nu) du = \frac{\sin{[m+n]u}}{2[m+n]} + \frac{\sin{[m-n]u}}{2[m-n]} + C
  • \int \cos(nu)\sin(mu) du = -\frac{\cos{[m+n]u}}{2[m+n]} - \frac{\cos{[m-n]u}}{2[m-n]} + C
  • \int u \sin{u} du = \sin{u} - u \cos{u} + C
  • \int u \cos{u} du = \cos{u} + u \sin{u} + C