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Chapter 5: Q 45. (page 452)

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

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

01

Step 1. Given information

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

02

Step 2. Explanation

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

Let $s e c x = u$

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

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

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Most popular questions from this chapter

Solve given definite integral.

${鈭�}_{0}^{4} 鈥� x^{3} \sqrt{x^{2} + 4} d x$

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x^{2} 鈭� 4 x 鈭� 8$

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$鈭� \frac{x}{\sqrt{x^{2} - 1}} d x$

Write $\sin (2 \cos^{- 1} x)$ as an algebraic function.

Problem Zero: Read the section and make your own summary of the material.

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