# Can this integral be done using elementary techniques?

1. Aug 20, 2013

### Bipolarity

1. The problem statement, all variables and given/known data

$\displaystyle \int \frac{dx}{(x^{2}+a^{2})^{3/2}}$

2. Relevant equations

3. The attempt at a solution
A substitution does not seem to work. I know that a closed form solution exists however, just curious if it can be done by the standard techniques usually taught in calculus. Thanks!

BiP

2. Aug 20, 2013

### Infrared

The substitution $x=asinh\theta$ seems to work.

Edit: $x=atan\theta$ also works if you don't mind integrating the secant function.

Last edited: Aug 20, 2013
3. Aug 20, 2013

### Goa'uld

Actually, you need to integrate cosine after that substitution. It may be the easiest way.

4. Aug 20, 2013

### Infrared

You are right of course. Somehow I turned $\frac{1}{sec\theta}$ into $\frac{1}{cos\theta}$

5. Aug 20, 2013

### Staff: Mentor

The presence of x2 + a2 in the denominator makes this integral a candidate for trig substitution. Same goes for the difference of squares, either x2 - a2 or a2 - x2.

What I do in these situations is draw a right triangle, and label the sides and hypotenuse according to whether I'm dealing with a sum of squares or a difference. If it's a sum of squares, as in this problem, I label the vertical side as x and the horizontal side as a. This means that the hypotenuse is √(x2 + a2). If θ is the angle across from x, then tanθ = x/a, so sec2θdθ = dx/a, or dx = asec2θdθ. After this, replace x and dx in the integral with θ and dθ and integrate.

A similar analysis can be done for either of the two differences of squares.