Substitution Rule for Integrals: How to Simplify Complex Integrands

[Arnoldjavs3]

Homework Statement

$$\int_{0}^{2} r\sqrt{5-\sqrt{4-r^2}} dr$$

Homework Equations

The Attempt at a Solution

would i substitute $u=4-r^2$?

After of which I would input into the integral and get:
$$\int_{0}^{2} \sqrt{5-\sqrt{u}}du$$

What would I do here? Do I just work inside the radical(so 5r - 2/3u^3/2)? Or do I have to account that it's inside a radical when solving?

 

[Mark44]

Homework Statement

$$\int_{0}^{2} r\sqrt{5-\sqrt{4-r^2}} dr$$

Homework Equations

The Attempt at a Solution

would i substitute $u=4-r^2$?

After of which I would input into the integral and get:
$$\int_{0}^{2} \sqrt{5-\sqrt{u}}du$$

This doesn't look very promising, and besides, it doesn't look like you did your substitution correctly. If $u = 4 - r^2$, what is du? You can't simply replace dr by du.

What would I do here? Do I just work inside the radical(so 5r - 2/3u^3/2)? Or do I have to account that it's inside a radical when solving?

As I said, this doesn't seem very promising. I would try this substitution: $u = \sqrt{4 - r^2}$, or equivalently, $u^2 = 4 - r^2$.

 

[alan2]

Substitute $$u = 5 - \sqrt {4 - {r^2}}$$. Then $$rdr = (5 - u)du$$ and your integral becomes

$$\int_3^5 {(5 - u)\sqrt u du} $$

I didn't actually evaluate it because now it's trivial.