Mathematics.

integral calculus

Trigonometric Substitution

Calculus II20 minDifficulty6 out of 10

You should know: integral, trigonometric functions

Overview

Trigonometric substitution replaces an algebraic expression involving √(a²-x²), √(a²+x²), or √(x²-a²) with a trigonometric function, exploiting the Pythagorean identities to eliminate the square root and simplify the integral.

Interactive Graph

sqrt(1-x^2) — the classic trig-substitution shape

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Formal Definition

Definition

The three standard substitutions, chosen based on the form under the radical:

a2−x2:x=asin⁡θ, dx=acos⁡θ dθ, a2−x2=acos⁡θ\sqrt{a^2-x^2}: \quad x = a\sin\theta,\ dx = a\cos\theta\,d\theta,\ \sqrt{a^2-x^2}=a\cos\theta
Case 1
a2+x2:x=atan⁡θ, dx=asec⁡2θ dθ, a2+x2=asec⁡θ\sqrt{a^2+x^2}: \quad x = a\tan\theta,\ dx = a\sec^2\theta\,d\theta,\ \sqrt{a^2+x^2}=a\sec\theta
Case 2
x2−a2:x=asec⁡θ, dx=asec⁡θtan⁡θ dθ, x2−a2=atan⁡θ\sqrt{x^2-a^2}: \quad x = a\sec\theta,\ dx = a\sec\theta\tan\theta\,d\theta,\ \sqrt{x^2-a^2}=a\tan\theta
Case 3

Worked Examples

  1. This matches the √(a²-x²) case with a=3. Substitute x=3sin(θ), dx=3cos(θ)dθ.

    x=3sin⁡θ, 9−x2=3cos⁡θx=3\sin\theta,\ \sqrt{9-x^2}=3\cos\theta

  2. Substitute into the integral; the 3cos(θ) terms cancel.

    ∫3cos⁡θ dθ3cos⁡θ=∫dθ=θ+C\int \frac{3\cos\theta\,d\theta}{3\cos\theta} = \int d\theta = \theta + C

  3. Convert back to x using θ=arcsin(x/3).

    =arcsin⁡(x3)+C= \arcsin\left(\frac{x}{3}\right) + C

Answer: arcsin(x/3) + C

Practice Problems

Difficulty 6/10

Evaluate ∫ dx/√(x²+4).

Summary

  • Trigonometric substitution handles integrands with √(a²-x²), √(a²+x²), or √(x²-a²) by substituting x=a sinθ, a tanθ, or a secθ respectively.
  • Each substitution uses a Pythagorean identity to collapse the square root into a single trig function.
  • After integrating in θ, convert back to x using a reference right triangle built from the substitution.

References

  1. WebsiteWikipedia — Trigonometric substitution