---
title: "Trigonometric Integrals and Trigonometric Substitution"
description: "Handle powers of sine and cosine with identities, and clear expressions such as √(a² - x²) by substituting a trigonometric function for x."
canonical: https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg
source: https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg.md
retrieved: 2026-09-12
---

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# Trigonometric Integrals and Trigonometric Substitution

Handle powers of sine and cosine with identities, and clear expressions such as √(a² - x²) by substituting a trigonometric function for x.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg

## Ready when they can

- Reduce an odd power of sine by splitting off one factor and substituting
- Choose the substitution that matches a sum or difference of squares
- Convert the answer back to the original variable using a right triangle

## Lesson: Tame trig powers and rooted expressions

Powers of sine and cosine fall to one identity: sin^2 x + cos^2 x = 1. When a power is odd, split off a single factor to serve as du, convert the rest with the identity, and substitute. When the powers are even, half-angle formulas lower the degree instead.

**Example.** Take the integral of sin^3 x dx. Split it as sin^2 x sin x and convert: (1 - cos^2 x) sin x. Set u = cos x, so du = -sin x dx. The integral becomes an easy polynomial in u. For cos^5 x you mirror it: split off one cos x and set u = sin x.

Roots of squares invite a trig substitution, and the root shape picks it. Use x = a sin theta for sqrt(a^2 - x^2), x = a tan theta for sqrt(a^2 + x^2), and x = a sec theta for sqrt(x^2 - a^2). Convert dx too, such as dx = a cos theta d theta, integrate in theta, then draw a right triangle to return to x.

**Tip.** Matching the root shape is the whole decision: sqrt(16 - x^2) wants x = 4 sin theta. If theta stands alone at the end, write it as sin^-1(x/a). Never skip the trip back to x.

**Recap.** Split odd powers, halve even ones, and match each root to its trig substitution.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Hyperbolic Functions and Their Inverses](https://lightmysky.com/learn/mathematics/hyperbolic-functions-and-their-inverses-mt_gCUl1sWi9M)
- [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh)
- [Integration by Parts](https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf)

## Opens up

- [Partial Fractions for Rational Integrands](https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c)