Integration Using Inverse Trigonometric Functions

Integrals Whose Antiderivatives are Inverse Trigonometric Functions

These are some of the most important integration formulas to recognize. They arise from the derivatives of sin⁻¹(x), tan⁻¹(x), and sec⁻¹(x).

The following diagrams show some examples of Integration Rules that involve Inverse Trigonometric Functions. Scroll down the page for more examples and solutions on how to integrate using inverse trigonometric functions.

 

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Let a be a positive constant and u be a function of x (so du is its differential).

1. For Inverse Sine (arcsin):
   If your integrand looks like \(\frac{1}{\sqrt{a^2 - u^2}}\), the integral will involve sin⁻¹.
   \(\int{\frac{du}{\sqrt{a^2 - u^2}}} = sin^{-1}(\frac{u}{a}) + C\)

2. For Inverse Tangent (arctan):
   If your integrand looks like \(\frac{1}{\sqrt{a^2 + u^2}}\), the integral will involve tan⁻¹.
   \(\int{\frac{du}{a^2 + u^2}} = \frac{1}{a}tan^{-1}(\frac{u}{a}) + C\)

3. For Inverse Secant (arcsec):
   If your integrand looks like \(\frac{1}{u\sqrt{a^2 - u^2}}\), the integral will involve sec⁻¹.
   \(\int{\frac{du}{u\sqrt{u^2 - a^2}}} = \frac{1}{a}sec^{-1}(\frac{|u|}{a}) + C\)

Integration Involving Inverse Trig Functions - Part 1
The Basics
How to integrate involving inverse trigonometric functions?
Part 1 of 3

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Integration Involving Inverse Trig Functions - Part 2
Using Substitution
How to integrate involving inverse trigonometric functions?
Part 2 of 3

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Integration Involving Inverse Trig Functions - Part 3
Definite Integration and Completing the Square
How to apply the integrations rules involving inverse trigonometric function?
Part 3 of 3

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Integration Using Inverse Trigonometric Functions - Ex 1
This video gives two formulas and shows how to solve a problem with a bit of algebra and a u-substitution.

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Integration Using Inverse Trigonometric Functions - Ex 2
This video gives two formulas and shows how to solve a definite integral using u-substitution and the formulas.

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Trigonometric integrals - using inverse trig functions

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Integrals that result in inverse trigonometric functions

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