The integral of (sin x)^n

You can use parts to derive recursion formulas which express integrals in terms of simpler integrals.

I'll use parts to derive a recursion formula for the integral of (sin x)^n, where n > 1. I'll put u = (sin x)^(n-1) in the derivative column and dv = sin x dx in the integral column.

Now I'll take the equation and do some algebra.

Here's how I did each step.

1. I applied the identity (cos x)^2 = 1 - (sin x)^2.
2. In the integral on the right, I multiplied (sin x)^(n-2) by 1 - (sin x)^2; then I broke the integral up into two integrals.
3. The second integral on the right is a copy of the original integral. I moved it to the left side of the equation by adding (n - 1) times the original integral to both sides.
4. Finally, I divided both sides of the equation by n to get the recursion formula.

You can use the recursion formula to integrate powers of sin x. The formula replaces the integral with some junk, plus another integral of sin x with a smaller power. You can apply the recursion formula to the new sin x integral; if you keep going, the power eventually goes down to 0 or 1, at which point you can do the integral directly.

Here's how this looks with n = 3.

To integrate (sin x)^3, I replaced all the n's in the recursion formula with "3". I obtained a new integral, the integral of sin x, which I computed using a basic formula.

• There are other ways to integrate powers of sin x. For example, even powers of sin x can be simplified using the double angle formulas. For small powers, alternative methods may be easier.
• You shouldn't memorize recursion formulas like this one.
  • Remember that these formulas exist, and look them up in tables of integrals if you need to use one.
  • Remember that you can derive recursion formulas using integration by parts.

• Introduction: Integration by parts
• How do you set up an integration by parts table?
• The integral of x^2 e^x
  • How do you choose what to put in the table?
• The integral of ln x
• The integral of e^(2x) cos (3x)
• A recursion formula for the integral of (sin x)^n
• Computing a definite integral using integration by parts

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Last updated: June 13, 2005

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