Example of Integrals Involving Trig Functions

Example: Estimate the following integral.

∫ sin⁵ x dx

Solution

This integral no longer contains the cosine in it that would permit us to make use of the substitution that we used above.  Hence, that substitution won't work and we are going to have to find out other way of doing this kind of integral.

Let us first notice that we could write the integral like this,

∫ sin⁵ x dx = ∫ sin⁴ x sin x dx = ∫ (sin² x)² sin x dx

Here we recall the trig identity,

cos² x + sin² x = 1

⇒         sin² x = 1 - cos² x

Along with this identity the integral can be written as,

∫ sin⁵ x dx = ∫ (1- cos²x)² sin x dx

and we can now make use of the substitution u = cos x .  Doing this gives us,

∫ sin⁵ x dx = -∫ (1 - u²)² du

= -∫1 - 2u² + u⁴ du

= - (u - 2 u³ +1 u⁵) + c

= - cos x + 2 cos³ x - 1/5 cos⁵ x + c

Thus, with a little rewriting on the integrand we were capable to reduce this to a fairly simple substitution.