Mathematics.

integral calculus

Trigonometric Integrals

Calculus II20 minDifficulty5 out of 10

You should know: integral, trigonometric identities

Overview

Trigonometric integrals are integrals of products and powers of trigonometric functions, such as ∫sinᵐ(x)cosⁿ(x)dx. They're evaluated using trigonometric identities (Pythagorean, double-angle, and power-reduction formulas) to reduce the integrand to a form solvable by u-substitution.

Interactive Graph

sin^3(x) — the integrand

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Formal Definition

Definition

Key strategies for ∫sinᵐ(x)cosⁿ(x)dx:

If m is odd: save one sin⁡(x), convert the rest via sin⁡2x=1−cos⁡2x, substitute u=cos⁡x\text{If } m \text{ is odd: save one } \sin(x), \text{ convert the rest via } \sin^2 x = 1-\cos^2 x, \text{ substitute } u=\cos x
Odd power of sine
If n is odd: save one cos⁡(x), convert the rest via cos⁡2x=1−sin⁡2x, substitute u=sin⁡x\text{If } n \text{ is odd: save one } \cos(x), \text{ convert the rest via } \cos^2 x = 1-\sin^2 x, \text{ substitute } u=\sin x
Odd power of cosine
If both m,n are even: use sin⁡2x=1−cos⁡2x2, cos⁡2x=1+cos⁡2x2\text{If both } m,n \text{ are even: use } \sin^2 x = \frac{1-\cos 2x}{2},\ \cos^2 x = \frac{1+\cos 2x}{2}
Both powers even

Worked Examples

  1. Odd power of sine: save one factor of sin(x), rewrite the rest using sin²x=1-cos²x.

    ∫sin⁡3x dx=∫sin⁡2x⋅sin⁡x dx=∫(1−cos⁡2x)sin⁡x dx\int \sin^3 x\,dx = \int \sin^2 x \cdot \sin x\,dx = \int (1-\cos^2 x)\sin x\,dx

  2. Substitute u=cos(x), du=-sin(x)dx.

    =−∫(1−u2) du=−u+u33+C= -\int (1-u^2)\,du = -u + \frac{u^3}{3} + C

  3. Substitute back u=cos(x).

    =−cos⁡x+cos⁡3x3+C= -\cos x + \frac{\cos^3 x}{3} + C

Answer: -cos(x) + cos³(x)/3 + C

Practice Problems

Difficulty 5/10

Evaluate ∫ cos³(x) dx.

Summary

  • Trigonometric integrals of the form ∫sinᵐx cosⁿx dx are solved via Pythagorean identity substitution when an odd power is present.
  • When both powers are even, power-reduction (half-angle) identities convert the integrand into a sum of cosines of multiple angles.
  • The general strategy always aims to reduce the integral to a u-substitution.

References

  1. WebsiteWikipedia — List of integrals of trigonometric functions