Question

How do you integrate `int sec^2(4x)(3-4tan(4x))^3 dx` by using the substitution method?

Answer 1

Let `u = 3 - 4tan(4x)", then "du = -16sec^2(4x)dx`
`sec^2(4x)dx = -1/16du`

Explanation:

Substitute the above into the integral

`intsec^2(4x)(3 - 4tan(4x))^3dx = -1/16intu^3du = -u^4/64 + C`

Reverse the substitution:

`intsec^2(4x)(3 - 4tan(4x))^3dx = -(3 - 4tan(4x))^4/64 + C`