# another root integral

• Feb 9th 2013, 10:56 PM
infraRed
another root integral
Hi. As usual, struggling with basic integration stuff I'm already supposed to have mastered. Could somebody point me in the right direction with the following integral? I'm thinking maybe it is a trigonometric type of problem, but that is just a guess. Played around with addition by 0 and multiply by 1 tricks, but nothing seems to have helped so far.

$\displaystyle \int{\sqrt{r^2-x^2}\,dx}$
• Feb 9th 2013, 11:15 PM
billb91
Re: another root integral
The form $\displaystyle \int \sqrt{a^2-x^2} \,dx$ is associated with trig substitution, specifically $\displaystyle x=a \sin \theta$. Hope that helps.
• Feb 10th 2013, 02:17 PM
infraRed
Re: another root integral
Quote:

Originally Posted by billb91
The form $\displaystyle \int \sqrt{a^2-x^2} \,dx$ is associated with trig substitution, specifically $\displaystyle x=a \sin \theta$. Hope that helps.

Thanks. If I might ask: what integral table do you use, or how did you know that? I can't seem to find that in my integral table.
• Feb 10th 2013, 03:19 PM
billb91
Re: another root integral
Honestly, when we went over trig substitution in my class, my teacher told us that secant, sine, and tangent were the three main ones we needed to know, and if we ever saw integrals in the forms:

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

or

$\displaystyle \int \sqrt {x^2-a^2} \,dx$

then the corresponding trig functions would be tangent (1) and secant (2), while I already showed you sine. Makes sense, as

$\displaystyle \int \frac {1}{\sqrt {1-x^2}} \,dx$ = $\displaystyle \arcsin x+c$
$\displaystyle \int \frac {1}{x\sqrt {x^2-1}} \,dx$ = $\displaystyle sec^-1 x+c$
$\displaystyle \int \frac {1}{{x^2+1}} \,dx$ = $\displaystyle \arctan x+c$

So you can see the similarities in the denominators.