← Calculus II

Unit 5: Trigonometric Substitution

A square root that won't simplify meets a right triangle, and the whole integral turns into trigonometry.

4 topics · 16 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

The Sine Substitution for √(a2 - x2)

  1. How it worksHow does drawing a right triangle with hypotenuse a and opposite side x turn x = a sin(theta) into a simplification of √(a2 - x2)?
  2. How it worksHow does labeling the triangle's adjacent side a cos(theta) let you read every leftover piece straight off the picture?
  3. Worked exampleFind the integral of 1/√(9 - x2) dx using the substitution x = 3 sin(theta).
  4. Common mistakeWhy doesn't ordinary u-substitution ever remove the square root in √(a2 - x2)?
  5. Going furtherWhat happens to the triangle and the substitution when the square root is √(1 - 4x2) instead?
Your turn1 of 2
Using the substitution x = √7 sin(theta), find F(x), the antiderivative of f(x) = 4/√(7 - x2), satisfying F(0) = 0.

The Tangent Substitution for √(a2 + x2)

  1. How it worksHow does drawing a right triangle with legs a and x and setting x = a tan(theta) turn √(a2 + x2) into a sec(theta)?
  2. Worked exampleFind the integral of 1/√(4 + x2) dx using the substitution x = 2 tan(theta).
  3. Common mistakeWhy isn't the tangent substitution just the sine substitution with a different letter?
  4. Going furtherWhat happens to the final answer once sec(theta) and tan(theta) have to convert back into x from the triangle?
Your turn1 of 2
Using the substitution x = √4 tan(theta), find F(x), the antiderivative of f(x) = 4/√(4 + x2), satisfying F(0) = 0.

The Secant Substitution for √(x2 - a2)

  1. How it worksHow does setting x = a sec(theta) and drawing the triangle with hypotenuse x turn √(x2 - a2) into a tan(theta)?
  2. Worked exampleFind the integral of 1/√(x2 - 16) dx using the substitution x = 4 sec(theta).
  3. Common mistakeWhy doesn't the secant substitution work when x is smaller than a?
Your turn1 of 2
Using the substitution x = √3 sec(theta), find F(x), the antiderivative of f(x) = 4/√(x2 - 3), satisfying F(4) = 0.

Completing the Square Before a Trig Substitution

  1. How it worksHow does completing the square turn x2 + 6x + 13 into (x + 3)2 + 4, a shape a trig substitution can use?
  2. Worked exampleFind the integral of 1/√(x2 + 6x + 13) dx by completing the square first.
  3. Common mistakeWhy doesn't a quadratic under a square root automatically match one of the three trig substitution patterns?
  4. Going furtherWhat happens to the triangle's labels once the substitution is written in terms of x + 3 instead of x?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 1/√(x2 + 6x + 13), satisfying F(-3) = 0.