Indefinite Integral?

Question

Integral: 1/(x^2(sqroot(144-x^2))dx

Answer 1

∫1/(x²√(144-x²)) dx

First, extract the factor of 1/12:

1/12 ∫1/(x²√(1-(x/12)²)) dx

Now, let u=arcsin (x/12), so that x=12 sin u, dx = 12 cos u du:

∫1/(144 sin² u √(1-sin² u)) cos u du

Simplifying:

1/144 ∫cos u/(sin² u √(1-sin² u)) du
1/144 ∫cos u/(sin² u √cos² u) du
1/144 ∫cos u/(sin² u cos u) du
1/144 ∫1/(sin² u) du
1/144 ∫csc² u du

Now integrate:

-1/144 cot u + C

Now consider that cot² u = csc² u-1 = 1/sin² u-1 = 144/x²-1 = (144-x²)/x², so cot u = ±√(144-x²)/x. As for the sign, consider that whenever x>0 ↔ u>0 ↔ cot u>0, so the sign of cot is the same as the sign of x, which is true if we take the positive sign in the above expression, i.e. cot u = √(144-x²)/x. So the integral is:

-√(144-x²)/(144x) + C

And we are done.