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Chapter 4: Q. 65 (page 363)

Solve each of the integrals in Exercises 63鈥�68, where a, b, and c are real numbers with
$a 鈮� 0, b 鈮� 0, c > 1, a n d c 鈮� 0 . 鈭� \frac{\sec (a x) \tan (a x)}{b} d x .$

Short Answer

Expert verified

The value of the given integral is $\frac{1}{a b} \sec (a x) + c .$

Step by step solution

01

Step 1. Given Infromation

Given is the integral:

$鈭� \frac{\sec (a x) \tan (a x)}{b} d x, a 鈮� 0, b 鈮� 0, c > 1, a n d c 鈮� 0, a n d a, b, a n d c a r e c o n s \tan t s .$

02

Step 2. Formula involved.

$鈭� s e c (x) \tan (x) d x = s e c (x) + c .$

03

Step 3. Solving the integral.

$鈭� \frac{s e c (a x) \tan (a x)}{b} d x l e t t = a x d t = a d x P u t t i n g t h e v a l u e i n t h e i n t e g r a l = \frac{1}{b} 鈭� \frac{s e c (t) \tan (t) d t}{a} = \frac{1}{a b} 鈭� s e c (t) \tan (t) d t = \frac{1}{a b} s e c t + c = \frac{1}{a b} s e c (a x) + c .$

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Most popular questions from this chapter

Suppose f is a function whose average value on $[- 3, 1]$ is

$- 2$ and whose average rate of change on the same in-

terval is $4$ . Sketch a possible graph for f . Illustrate the

average value and the average rate of change on your

graph of f .

Given a simple proof that ${鈭�}_{k = 0}^{n} 3 a_{k} = 3 {鈭�}_{k = 0}^{n} a_{k}$

Given formula for the areas of each of the following geometric figures

a) area of circle with radius r

b) a semicircle of radius r

c) a right triangle with legs of lengths a and b

d) a triangle with base b and altitude h

e) a rectangle with sides of lengths w and l

f) a trapezoid with width w and height $h_{1}, h_{2}$

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{0}^{1} 鈥� \frac{1}{1 + x^{2}} d x$

Find the sum or quantity without completely expanding or calculating any sums.

Given ${鈭�}_{k = 1}^{4} a_{k} = 7$ and $, {鈭�}_{k = 0}^{4} b_{k} = 10$ , $a_{0} = 2$ . Find the value of ${鈭�}_{k = 0}^{4} (2 a_{k} + 3 b_{k})$ .

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