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Chapter 5: Q 16. (page 451)

Describe strategy for solving the type of integral given.
$\int s e c^{m} x \tan^{n} x d x, m$ even

Short Answer

Expert verified

Use the formula: $s e c^{2} x = 1 + \tan^{2} x$

Step by step solution

Step 1. Given information

Integral is $\int s e c^{m} x \tan^{n} x d x, m$ even

Step 2. Explanation

Consider the integral $\int s e c^{m} x \tan^{n} x d x, m$ even

Write $s e c^{m} x = s e c^{2} x s e c^{m - 2} x$ , then use the formula $s e c^{2} x = 1 + \tan^{2} x$

Then let $\tan x = u$ and use the formula $(\tan x)' = s e c^{2} x$

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Most popular questions from this chapter

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

$\int x \sqrt{x^{2} + 1} d x$

Solve the integral

$\int \frac{1}{\sqrt{x^{2} + 9}} d x$

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

$\int e^{u} d u$

Suppose $u (x) = x^{2}$ . Calculate and compare the values of the following definite integrals:

role="math" localid="1648786835678" $\int_{- 1}^{5} u^{2} d u, \int_{x = - 1}^{x = 5} u^{2} d u and \int_{u (- 1)}^{u (5)} u^{2} d u$

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

$x^{2} - 4 x - 8$

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Describe strategy for solving the type of integral given.∫secmxtannxdx,meven

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Describe strategy for solving the type of integral given.
$\int s e c^{m} x \tan^{n} x d x, m$ even