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$$\theta$$d$$\theta$$
$$\sqrt{4x^{2}+9}$$=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
$$\int$$cot$$\theta$$csc³$$\theta$$d$$\theta$$
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?