Integration with square roots in the denominator

**chengbin** · June 12th 2009, 12:22 PM

$\int \frac {dx}{\sqrt {x^2+a^2}}$

In my solution book it says

Let $t=x+\sqrt {x^2+a^2}$

How do you figure what to use as substitution

I'm too lazy to type this out, so here's a picture.

In the circled part, how do they get from that trig expression to that?

**Amer** · June 12th 2009, 12:32 PM

$\left( \frac{1+sinx}{cosx}\right)^2=\left(\frac{1}{cosx} +\frac{sinx}{cosx}\right)$

for your integration use the sub

let $atan(u)=x$ ..... $asec^2(u) du=dx$

$tanu=\frac{x}{a}$ as you can see in the attachment

$sec(u)=\frac{\sqrt{a^2+x^2}}{a}$ the rest for you

Integration with square roots in the denominator-sesesese.jpg

**Plato** · June 12th 2009, 12:40 PM

Go this website
Type in the exact expression: integrate dx/sqrt[x^2 + a^2]
Click the equal sign at the end. What do you get?
Ckick the show steps button.