Evaluating difficult integral involving square roots

[hali]

Homework Statement

Evaluate the following integral

Homework Equations

∫ √(4-(√x)) dx

The Attempt at a Solution

I am having a mind block, I find this too challenging, help!

 

[HallsofIvy]

Well, the instant I look at that I think of setting $u= 4- \sqrt{x}= 4- x^{1/2}$. Then $du= -(1/2)x^{-1/2}dx$ so that $dx= -2x^{1/2}du= -2\sqrt{x}du$.

And, since $u= 4- \sqrt{x}$, $\sqrt{x}= 4- u$.

 

[hali]

I understand the du = ... But what happens to the square root of the entire function?

 

[scurty]

I understand the du = ... But what happens to the square root of the entire function?

That's the whole point of setting $u = 4 - \sqrt{x}$, the integral then becomes $\int \sqrt{u} \ (2u-8) \ du$, you simply substitute u in for $4 - \sqrt{x}$.

There are other ways of solving this integral as well. You can use trig substituion and you can also set $u^2 = 4 - \sqrt{x}$ and still arrive at the correct result.