# Integration by trigonometric substitution

## 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.

## Answers and Replies

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Defennder
Homework Helper
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?

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