← Calculus II

Unit 4: Trigonometric Integrals

Powers and products of sine, cosine, tangent, and secant, cracked open by identity instead of raw substitution.

5 topics · 22 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.

Powers of Sine and Cosine with an Odd Exponent

  1. How it worksHow does peeling off a single sine factor and converting the rest to cosine set up a clean u-substitution?
  2. Common mistakeWhy doesn't splitting sin3(x) into sin(x) times sin2(x) do anything without the Pythagorean identity too?
  3. Worked exampleFind the integral of sin3(x) cos2(x) dx.
  4. Worked exampleFind the integral of sin5(x) dx.
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4sin3(x)cos2(x), satisfying F(0) = 0.

Powers of Sine and Cosine with Only Even Exponents

  1. How it worksHow do the power-reduction identities trade sin2(x) and cos2(x) for cos(2x) terms you can integrate?
  2. Worked exampleFind the integral of sin2(x) dx using the power-reduction identity.
  3. Worked exampleFind the integral of cos4(x) dx using power reduction twice.
  4. Common mistakeWhy doesn't u-substitution with u = sin(x) work on an integral of cos2(x) alone?
  5. Going furtherWhat happens to the power-reduction approach when sin2(x) and cos2(x) appear together with even exponents?
Your turn1 of 2
Find the definite integral from 0 to π of 4 sin2(x) dx.

Integrals of Powers of Tangent and Secant

  1. How it worksHow does the identity 1 + tan2(x) = sec2(x) let you peel off a sec2(x) du and substitute?
  2. Common mistakeWhy doesn't the integral of tan(x) dx come out to sec2(x) the way the derivative rule might suggest?
  3. Worked exampleFind the integral of tan3(x) sec2(x) dx.
  4. Worked exampleFind the integral of tan2(x) sec4(x) dx, using an even power of secant.
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4tan3(x)sec2(x), satisfying F(0) = 0.

Products of Sines and Cosines with Different Periods

  1. How it worksHow do the product-to-sum identities turn a product of two different-frequency waves into a sum you can integrate term by term?
  2. Common mistakeWhy isn't the integral of sin(3x)cos(5x) dx just the product of the two separate antiderivatives?
  3. Worked exampleFind the integral of sin(2x) cos(5x) dx using a product-to-sum identity.
  4. Worked exampleFind the integral of cos(4x) cos(x) dx using a product-to-sum identity.
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4sin(2x)cos(5x), satisfying F(0) = 0.

Choosing Which Identity to Reach For

  1. How it worksHow do you decide between power reduction, a Pythagorean swap, and a product-to-sum identity just from the exponents in front of you?
  2. Worked exampleFind the integral of sin2(x) cos3(x) dx by picking the identity that fits the odd cosine power.
  3. Worked exampleFind the integral of tan4(x) dx by picking the identity that fits an all-tangent integrand.
  4. Common mistakeWhy doesn't memorizing one favorite trig identity get you through every trig integral?
  5. Going furtherWhat happens when an integral mixes tangent, secant, sine, and cosine all in one expression?
Your turn1 of 2
Which identity or technique is the right first move for the integral of sin2(x)cos3(x) dx?