Question

An Integral problem:
(1/((9x^2)-16)^(1/2)
Can't figure out where to start.

Answer 1

It involves a trigonometric substitution:
\[\int\frac{1}{\sqrt{9x^2-16}}~dx\]
Let \(x=\dfrac{4}{3}\sec u~\Rightarrow~dx=\dfrac{4}{3}\sec u\tan u~du\).
\[{\Large\int}\frac{1}{\sqrt{9\left(\dfrac{4}{3}\sec u\right)^2-16}}~\left(\dfrac{4}{3}\sec u\tan u~du\right)\\
\frac{4}{3}\int\frac{\sec u\tan u}{\sqrt{16\sec^2 u-16}}~du\\
\frac{4}{3}\int\frac{\sec u\tan u}{\sqrt{16}\sqrt{\sec^2 u-1}}~du\\
\frac{4}{3\sqrt{16}}\int\frac{\sec u\tan u}{\sqrt{\tan^2u}}~du\\
\frac{1}{3}\int\sec u~du\]

Answer 2

If you prefer the hyperbolic functions you might want to try

\[x=\frac{ 4 }{ 3 }\cosh \alpha\]
as a substitution too, however @SithsAndGiggles method is really better if you're allowed to use a table of integrals.

Answer 3

I see. Once trig is involved in these problems I work, all bets are off. Completely confuses. But this helps guys! Thanks!!

Answer 4

Ok, let me figure out how to give "thanks,' to you guys.