1. ## antiderrivative of...

Im having trouble finding the antiderrivative of r/[sqrt(4+r^2)]

the answer is [sqrt(4+r^2)]...but how can that be?

2. Originally Posted by johntuan
Im having trouble finding the antiderrivative of r/[sqrt(4+r^2)]

the answer is [sqrt(4+r^2)]...but how can that be?

$\int \frac{r}{\sqrt{4+r^2}}dr$

let $u=4+r^2 \implies du=2rdr \iff \frac{du}{2}=rdr$

$\int \frac{r}{\sqrt{4+r^2}}dr=\frac{1}{2}\int \frac{1}{u^{1/2}}du=...$

3. but then wouldnt the answer be over 2?

u^(1/2)/2

but the answer doesnt have a denominator:s

4. Not quite

$\frac{1}{2}\int \frac{1}{u^{1/2}}du=\frac{1}{2}\int u^{-1/2}du=\frac{\frac{1}{2}}{\frac{1}{2}}u^{1/2}=\sqrt{u}=\sqrt{4+r^2}$