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.

 

Trigonometry Worksheets
Practice your skills with the following worksheets:
Printable & Online Trigonometry Worksheets

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

• Show Video Lesson

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

• Show Video Lesson

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

• Show Video Lesson

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.

• Show Video Lesson

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.

• Show Video Lesson

Trigonometric integrals - using inverse trig functions

• Show Video Lesson

Integrals that result in inverse trigonometric functions

• Show Video Lesson

Check out our most popular games!
Fraction Concoction	Fact Family Game	Number Bond Garden
Online Addition Subtraction Game	Penguin Solitaire	Sawayama Solitaire
Ark Solitaire	Eldritch Invasion	Shenzhen Solitaire

Check out more Solitaire games here.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.