What is the answer to this trig sub integral: ∫ (x+1)^2 / (1- x^2)^(3/2) dx???

Question

I think you have to use trig substitution, but i have never seen a question like this before. A fully supported answer would be much appreciated.
thanks

Answer 1

Let x = sin u, then dx=cos u du. So we have:

∫(sin u+1)²/(1-sin² u)^(3/2) cos u du

Using the Pythagorean theorem:

∫(sin u+1)²/(cos² u)^(3/2) cos u du

Computing the power and simplifying:

∫(sin u+1)²/cos² u du

Expanding the numerator:

∫(sin² u + 2 sin u + 1)/(cos² u) du

Expanding further:

∫tan² u + 2 sec u tan u + sec² u du

Now, tan² u = sec² u - 1, so:

∫2 sec² u + 2 sec u tan u - 1 du

All of these are derivatives you should know:

2 tan u + 2 sec u - u + C

Now, expanding:

2 sin u/cos u + 2/cos u - u + C

Rewriting the cosines:

2 sin u/√(1-sin² u) + 2/√(1-sin² u) - u + C

And resubstituting:

2x/√(1-x²) + 2/√(1-x²) - arcsin x + C

And we are done.

Answer 2

oo thats a pretty symbol, howd you do that?