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Chapter 4: Q. 63 (page 363)

Solve each of the integrals in Exercises 63鈥�68, where a, b, and c are real numbers with
$a 鈮� 0, b 鈮� 0, c > 1, a n d c 鈮� 0 .$
$鈭� a e^{b x + c} d x .$

Short Answer

Expert verified

The value of the given integral is $\frac{a e^{b x + c}}{b} + k, w h e r e k i s a c o n s \tan t .$

Step by step solution

01

Step 1. Given Information.

Given is a integral: $鈭� a e^{b x + c} d x,$

$a 鈮� 0, b 鈮� 0, c > 1, a n d c 鈮� 0, a n d a, b, a n d c a r e c o n s \tan t s .$

02

Step 2. Formula involved.

$鈭� e^{b x} d x = \frac{e^{b x}}{b} + k, 鈭� a e^{x} d x = a e^{x} + k .$

03

Step 3. Solving the integral.

$鈭� a e^{b x + c} d x = a 鈭� e^{b x + c} d x l e t t = b x + c d t = b d x P u t t i n g i n t h e i n t e g r a l w e g e t, = \frac{a}{b} 鈭� e^{t} d t = \frac{a}{b} e^{t} + k = \frac{a}{b} e^{b x + c} + k .$

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Most popular questions from this chapter

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

${鈭�}_{0}^{蟺} 鈥� (\sin^{2} 鈦� x + \cos^{2} 鈦� x) d x$

What is the difference between an antiderivative of a function and the indefinite integral of a function?

Your calculator should be able to approximate the area between a graph and the x-axis. Determine how to do this on your particular calculator, and then, in Exercises 21鈥�26, use the method to approximate the signed area between the graph of each function f and the x-axis on the given interval [a, b].

$f (x) = 1 - 2^{x}, [a, b] = [- 3, 1]$

Read the section and make your own summary of the material.

Consider the region between f and g on [0, 4] as in the

graph next at the left. (a) Draw the rectangles of the left-

sum approximation for the area of this region, with n = 8.

Then (b) express the area of the region with definite

integrals that do not involve absolute values.

See all solutions

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