Trigonometric integrals

Definition

Trigonometric integrals are integrals that involve trigonometric functions such as sine, cosine, and tangent. These integrals often require specific techniques for simplification and evaluation.

5 Must Know Facts For Your Next Test

1. To solve trigonometric integrals involving $\sin^m(x) \cos^n(x)$, use trigonometric identities to simplify the expression.
2. For integrals like $\tan^m(x) \sec^n(x)$, substitution using $u = \sec(x)$ or $u = \tan(x)$ can be helpful.
3. When integrating functions like $\sin(ax)\cos(bx)$, employ product-to-sum formulas to simplify the integral.
4. In cases where powers of sine and cosine are both even, use half-angle identities to reduce the powers.
5. The integral of a secant function often involves a natural logarithm, specifically $\int \sec(x)dx = \ln |\sec(x) + \tan(x)| + C$.

Review Questions

• How would you approach integrating $\sin^3(x) \cos^2(x)$?
• What substitution can you use for the integral of $\tan^4(x) \sec^2(x)$?
• Explain how to use product-to-sum formulas in simplifying the integral of $\sin(3x)\cos(5x)$.

Related terms

Trigonometric Substitution: A technique used to evaluate integrals by substituting trigonometric functions for algebraic expressions.

Partial Fractions: A method used to decompose rational functions into simpler fractions that are easier to integrate.

$u$-Substitution: A technique where one substitutes part of the integral with a new variable to simplify integration.