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Chapter 5: Q. 75 (page 452)

Solve each of the definite integrals in Exercises 67鈥�76.
${鈭�}_{0}^{蟺 / 4} \frac{\sin^{2} x}{\cos^{4} x} d x$ .

Short Answer

Expert verified

The answer is $\frac{1}{3}$ .

Step by step solution

01

Step 1. Given Information.

The integral is ${鈭�}_{0}^{蟺 / 4} \frac{\sin^{2} x}{\cos^{4} x} d x$ .

02

Step 2. Explanation.

Simplify the integral using trigonometric identities, $\tan x = \frac{\sin x}{\cos x}$ and $s e c x = \frac{1}{\cos x}$ .

${鈭�}_{0}^{蟺 / 4} \tan^{2} x s e c^{2} x d x$

Substitute $u = \tan x$ so, $d u = s e c^{2} x d x$

so, limits changes 0 to 1.

role="math" localid="1649243192222" ${鈭�}_{0}^{1} u^{2} d u$ .

03

Step 3. Calculation.

Further simplify.

${鈭�}_{0}^{1} u^{2} d u = {[\frac{u^{3}}{3}]}_{0}^{1} = \frac{1}{3}$

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Most popular questions from this chapter

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.

$鈭� \frac{\sqrt{4 - x^{2}}}{x^{2}} d x$

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

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

Find three integrals in Exercises 27鈥�70 for which either algebra or u-substitution is a better strategy than integration by parts.

For each function u(x) in Exercises 9鈥�12, write the differential du in terms of the differential dx.

$u (x) = x^{2} + 1$

Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.

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