Integration by trigonometric substitution

[cmajor47]

Homework Statement

Solve the integral using trigonometric substitution.
\int\frac{\sqrt{4x^{2}+9}}{x^{4}}dx

Homework Equations

2x=3tan\theta
x=2/3 tan\theta
dx=2/3 sec²\thetad\theta
\sqrt{4x^{2}+9}=3sec\theta

The Attempt at a Solution

\frac{8}{9} \int \frac{cos^{2}\theta}{sin^{4}\theta} d\theta

From here, I don't know how to get the integral into a form that is easy to solve.

 

[Defennder]

You made an error. Your numerator should be cos theta, not cos^2 theta. Now what do you know about how to differentiate and integrate powers of csc theta?

 

[cmajor47]

I fixed the error and have gotten to
\intcot\thetacsc³\thetad\theta
But how do I get this to any easy to solve integral?

 

[Defennder]

Well as I hinted earlier, what is the derivative of csc^n theta?