# Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant...

## Question:

Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.)

{eq}\int \frac{x^3}{\sqrt{x^2 + 64}}\;dx,\;\;x=8\,tan\,\theta {/eq}

## Trigonometric Substitution:

Trigonometric substitution is mainly used to remove radicals from integrands so the integral can be evaluated.

For arguments of the form,

$$a^2 - u^2, \hspace{1 cm} \text{let } u = a \sin \theta$$ $$u^2 - a^2, \hspace{1 cm} \text{let } u = a \sec \theta$$ $$u^2 + a^2, \hspace{1 cm} \text{let } u = a \tan \theta.$$

Since we will be utilizing {eq}\tan \theta{/eq}, let's review a couple of items.

a) The Pythagorean identity {eq} \tan^2 \theta + 1 = \sec^2 \theta{/eq}.

b) The derivatives of secant and tangent.

$$\frac{d}{d\theta} \sec \theta = \sec \theta \tan \theta$$ $$\frac{d}{d\theta} \tan \theta = \sec^2 \theta$$