Question

For the following integral find an appropriate TRIGNOMETRIC SUBSTITUTION of the form x=f(t) to simplify the integral.

INT((4x^2-3)^1.5) dx x=?

Answer 1

I would try something like this:

Integral (4x^2 - 3)dx
= integral (3(4x^2/3 - 1)dx

let (4/3)x^2 = sin^2Ѳ
then (2/√3)x = sinѲ
x = (√3/2)sinѲ
dx/dѲ = (√3/2)cosѲ
dx = (√3/2)cosѲ dѲ

so your integral turns into

integral (3(sin^2Ѳ - 1)^1.5 (√3/2)cosѲ dѲ

However, I see a problem arising.
sin^2 Ѳ - 1 ≤ 0 because of the sine function and we cannot raise a negative value to an exponent of 1.5

Perhaps you can see how to get around that.