What Is the Best Method to Integrate dx/(x*sqrt(9+16x^2))?

[cal.queen92]

Homework Statement

indefinite integral: dx/(x*sqrt(9+16x^2))

Homework Equations

Trig. Substitutions or parts??

The Attempt at a Solution

I tried using integration by parts but its got pretty messy...it also resembles a tan trig substitution, but it's within a square root. I'm stumped and can't figure out where to start...

Can anyone help?

Thanks!

 

[HallsofIvy]

Don't use integration by parts. Rewrite the problem as
$$\int\frac{xdx}{x^2\sqrt{9+ 16x^2}}$$
and let $u= 9+ 16x^2$.

 

[SammyS]

Factor a 3 out of $\sqrt{9 +16x^2}$

$$\displaystyle \sqrt{9 +16x^2}=\sqrt{9\left(1+\frac{16x^2}{9}\right)}=3 \sqrt{1+\frac{16x^2}{9}}=3 \sqrt{1+\left( \frac{4x}{3} \right)^2 }$$

We know that 1+tan²(θ) = sec²(θ) , so let 4x/3 = tan(θ), (4/3)dx = sec²(θ)dθ.

(I'm slow at typing, so HallsofIvy responded while I was typing. He usually has better ideas than I do. Good luck!)

 

[Dickfore]

See this solution:

http://www.wolframalpha.com/input/?i=Integrate[1%2F%28x+Sqrt[9%2B16*x^2]%29%2Cx]

There is a button that shows you the steps as well.