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Chapter 5: Q 44. (page 452)

Find the integral.
$\int \tan^{2} 4 x s e c 4 x d x$

Short Answer

Expert verified

Answer is $\frac{- 1}{8} \ln (|\tan (2 x + \frac{π}{4})|) + \frac{1}{8} \tan 4 x s e c 4 x + C$

Step by step solution

01

Step 1. Given information

Integral is $\int \tan^{2} 4 x s e c 4 x d x$

02

Step 2. Explanation

Let $4 x = u$

4 d x = d u

$\int \tan^{2} 4 x s e c 4 x d x = \frac{1}{4} \int \tan^{2} u s e c u d u = \frac{1}{4} \int (s e c^{2} u - 1) s e c u d u = \frac{1}{4} \int s e c^{3} u - s e c u d u = \frac{- 1}{8} \ln (|\tan (\frac{u}{2} + \frac{π}{4})|) + \frac{1}{8} \tan u s e c u + C = \frac{- 1}{8} \ln (|\tan (2 x + \frac{π}{4})|) + \frac{1}{8} \tan 4 x s e c 4 x + C$

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Most popular questions from this chapter

Solve the integral: $\int \frac{x^{2}}{e^{x}} d x$

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{x^{2} - 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving $\int \frac{1}{x^{2} + 4} d x .$

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $\int {(x^{2} + 4)}^{- 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} - a^{2}}$ .

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

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.

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

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

Show that if $x = \tan u$ , then $d x = \sec^{2} u d u$ , in the following two ways: (a) by using implicit differentiation, thinking of $u$ as a function of $x$ , and (b) by thinking of $x$ as a function of $u$ .

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