Chapter 5: Q 45. (page 452)

Solve the following integral.
$\int s e c^{3} x \tan^{5} x d x$

Short Answer

Expert verified

Answer is $\frac{s e c^{7} x}{7} - \frac{2 s e c^{5} x}{5} + \frac{s e c^{3} x}{3} + C$

Step by step solution

Step 1. Given information

An integral is $\int s e c^{3} x \tan^{5} x d x$

Step 2. Explanation

$\int s e c^{3} x \tan^{5} x d x = \int s e c^{3} x \tan^{3} x (s e c^{2} x - 1) d x$

Let $s e c x = u$

$s e c x \tan x d x = d u$

$\begin{array}{rcl} \int s e c^{3} x \tan^{5} x d x & = & \int u^{2} {(u^{2} - 1)}^{2} d u \\ = & \int u^{6} - 2 u^{4} + u^{2} d u \\ = & \frac{u^{7}}{7} - \frac{2 u^{5}}{5} + \frac{u^{3}}{3} + C \\ = & \frac{s e c^{7} x}{7} - \frac{2 s e c^{5} x}{5} + \frac{s e c^{3} x}{3} + C \end{array}$

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Solve the following integral.
$\int s e c^{3} x \tan^{5} x d x$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Step by step solution

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Solve the following integral.
$\int s e c^{3} x \tan^{5} x d x$