Question

Find the integral from 0 to 2 of xsqrt(5-sqrt4-x^2)).
The hint said to use substitution of u=sqrt(4-x^2), and that I needed one more substitution, but I don't know how to do it.

Answer 1

Sorry I forgot the dx after the integral.

Answer 2

∫[0,2] x√(5-√(4-x^2)) dx
Let
u = 5-√(4-x^2)
du = x/√(4-x^2) dx
so, x dx = (5-u) du
and you now have
∫[3,5] √u (5-u) du

and I'm sure you can handle that, eh?

Answer 3

Wait, why is it the integral from 3-5 instead of 0-2?

Answer 4

because now we are using u, not x.
If, after integration on u, you substitute back into terms of x, then you would use the limits [0,2].

When x=0, u=3 and when x=2, u=5.