Subject: Calculus

List Of Integrals Of Inverse Trigonometric Functions

The following are integrals of inverse trigonometric functions and its combinations and variations.

The Page at a Glance

Integrals of the 6 inverse trigonometric functions

  • \int \sin^{-1}{u} du = \int \arcsin{u} du = u\sin^{-1}{u} + \sqrt{1 - u^2} + C = u\arcsin{u} + \sqrt{1 - u^2} + C
  • \int \cos^{-1}{u} du = \int \arccos{u} du = u\cos^{-1}{u} - \sqrt{1 - u^2} + C = u\arccos{u} - \sqrt{1 - u^2} + C
  • \int \tan^{-1}{u} du = \int \arctan{u} du = u\tan^{-1}{u} - \ln{|\sqrt{1 + u^2}|} + C = u\arctan{u} - \ln{|\sqrt{1 + u^2}|} + C
  • \int \cot^{-1}{u} du = \int arccot {u} du =ucot^{-1}{u} + \ln{|\sqrt{1 + u^2}|} + C = u arccot{u} + \ln{|\sqrt{1 + u^2}|} + C
  • \int \sec^{-1}{u} du = \int arcsec {u} du = u \sec^{-1}{u} - \ln{|u + \sqrt{u^2 - 1}|} + C = u \sec^{-1}{u} - \cosh^{-1}{u} + C
  • \int \csc^{-1}{u} du = \int arccsc {u} du = u \csc^{-1}{u} + \ln{|u + \sqrt{u^2 - 1}|} + C = u \csc^{-1}{u} + \cosh{-1}{u} + C

Integrals of the 6 inverse functions with constants in the parameter

  • \int \sin^{-1}(bu) du = \int \arcsin(bu) du = u\sin^{-1}(bu) + \frac{\sqrt{1 - b^2u^2}}{b} + C = u\arcsin(bu) + \frac{\sqrt{1 - b^2u^2}}{b} + C
  • \int \cos^{-1}(bu) du = \int \arccos(bu) du = u\cos^{-1}(bu) - \frac{\sqrt{1 - b^2u^2}}{b} + C = u\arccos(bu) - \frac{\sqrt{1 - b^2u^2}}{b} + C
  • \int \tan^{-1}(bu) du = \int \arctan(bu) du = u\tan^{-1}(bu) - \frac{\ln{|\sqrt{1 + b^2u^2}|}}{2b} + C = u\arctan(bu) - \frac{\ln{|\sqrt{1 + b^2u^2}|}}{2b} + C
  • \int \cot^{-1}(bu) du = \int arccot (bu) du =ucot^{-1}(bu) + \frac{\ln{|\sqrt{1 + b^{2}u^{2}}|}}{2b} + C = u arccot(bu) + \frac{\ln{|\sqrt{1 + b^{2}u^{2}}|}}{2b} + C
  • \int \sec^{-1}(bu) du = \int arcsec (bu) du = u \sec^{-1}(bu) - \frac{1}{a}\tanh^{-1}{\sqrt{1-\frac{1}{b^2u^2}}} + C
  • \int \csc^{-1}(bu) du = \int arccsc (bu) du = u \csc^{-1}(bu) + \frac{1}{a}\tanh^{-1}{\sqrt{1-\frac{1}{b^2u^2}}} + C

Note: b is a constant.