3.2 Trigonometric Integrals

[STARTS AT 32:37]

Instructor: Number 30, tangent cubed x secant to the seventh x-- this seems like a place we're going to end up using one of our identities. Let's go ahead and take a tangent squared off of that tangent cubed-- tangent squared x times tangent times secant to the seventh x. So let's go ahead and-- actually, we're going to pull off a secant. We're going to pull a secant off of that as well. 

So this will be tangent squared x, secant to the sixth x, and a secant x, tangent x. Converting that tangent squared into 1-- make sure I get it right this times-- secant squared x minus 1, secant to the sixth x, secant x, tangent x, dx. And this becomes-- there's our du. 

So our u is going to be secant x. And our du will then be secant x, tangent x, dx. So this is going to be-- I'm going to go ahead and use my u substitution here. This will be u to the 8 minus u to the sixth, du. That will be u to the 9 over 9, minus u to the 7 over 7, plus c, and secant to the ninth over 9 minus secant to the 7 over 7, plus c. 

[ENDS AT 34:38]