Solving an Integral with Homework Help

[Melawrghk]

Homework Statement

\int \frac{26 dx}{{(169x^2+1)}^2} <= the whole denominator is supposed to be squared...

The Attempt at a Solution

So I converted the thing in the denominator so that it has a square root:
\int \frac{26 dx}{{\sqrt{169x^2+1}}^4}

Looking at the denominator, I realized I should do an inverse substitution:
13x=tan(t)
dx=sec²(t)*dt/13

I subbed that into the equation before and got:
\int \frac{26 * sec^2(t) * dt}{13*sec^4(t)}

Simplifying which, I get:
2\int cos^2(t) dt

Then I tried doing integration by parts, but I got nowhere - I kept getting cos^2 again... Please help me, this question frustrates me. Thanks in advance!

(Sidenote: I finally got my formulas all pretty, yay!)

 

[Dick]

There's not really much profit into converting x^2 into sqrt(x)^4, is there? The rest of the general approach looks fine. You get sec^2/sec^4. So sure, cos^2(x). You just want to use a double angle formula cos(x)^2=(1+cos(2x))/2.

 

[Melawrghk]

Thanks! I know there isn't a point with the root, I'm just more used to seeing it like that. And thanks for the double angle formula. I always forget they exist... I guess it's time I memorize them :)