# Simple Integration by substitution

## Homework Statement

Find by letting $$U^2=(4 + x^2)$$the following $$\int_0^2\frac{x}{\sqrt{4 + x^2}}dx$$?
I can solve it by letting $$\mbox{x=2} tan(\theta)$$, But I want to be able to do it by substitution.

## The Attempt at a Solution

$$\frac{du}{dx}=\frac{d\sqrt{(4+x^2)}}{dx}=\frac{x}{\sqrt{4+x^2}}\mbox{, therefore du}=\frac{x}{u^\frac{1}{2}}\times dx\\$$ Therefore the integral is $$\int_{x=0}^{x=2}\frac{1}{u^\frac{1}{2}}du=$$0.26757, it should be $$2(\sqrt{2}-1)$$. Can you tell me where I went wrong. Thanks for the help.

Last edited:

## Answers and Replies

Dick
Science Advisor
Homework Helper
[$$\frac{du}{dx}=\frac{d\sqrt{(4+x^2)}}{dx}=\frac{x}{\sqrt{4+x^2}}\mbox{, therefore du}=\frac{x}{u^\frac{1}{2}}\times dx\\$$

What happened to the x in the above expression for du? Did you just drop it?

There is a mistake in the equation, it should be $$\int_0^2\frac{x}{\sqrt{4+x^2}}dx$$. And the answer I then get =.5351. And thanks for the quick reply.

Dick
Science Advisor
Homework Helper
Ah, ha. Then you want to evaluate the final integral between the u limits, not the x limits.

Dick
Science Advisor
Homework Helper
$$\frac{du}{dx}=\frac{d\sqrt{(4+x^2)}}{dx}=\frac{x}{\sqrt{4+x^2}}\mbox{, therefore du}=\frac{x}{u^\frac{1}{2}}\times dx\\$$

And something is going awry here. A sqrt(u) would be the 4th root of 4+x^2. You are being sloppy.

I know there is something awry in the calculation, maybe you can find what I did wrong. Since $$u^2=4+x^2$$ that implies $$u=\sqrt{4+x^2}$$ and $$\frac{du}{dx}$$ expression follows.

Last edited:
Dick
Science Advisor
Homework Helper
This shows du=(x*dx)/u. Not u^(1/2).

First try substituting u=4+x²⇒du=2xdx⇒(1/2)du=xdx for the substitution part
that should be a lot easier . you are still heading the right direction
(1/2)∫(1/(u^{1/2}))du= √u
I think you can take it from here

Dick
Science Advisor
Homework Helper
He can also do it with his substitution. He just needs to be more careful of exactly what that is.

Thanks for the help lads I at last got it out using my substitution.