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Q. 40

Expert-verified

Calculus

Found in: Page 373

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals.
Use a graph to check your answer.
$\int_{- 3}^{2} \frac{1}{(x + 5)^{2}} d x$

Ans: The exact value is, $\int_{- 3}^{2} \frac{1}{(x + 5)^{2}} d x = \frac{5}{14}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

given expression,

$\int_{- 3}^{2} \frac{1}{(x + 5)^{2}} d x$

Step 2. The objective is to determine the exact value of the definite integral.

The exact value is calculated as shown below,

$\begin{aligned} \int_{- 3}^{2} \frac{1}{(x + 5)^{2}} d x \\ = \int_{- 3}^{2} (x + 5)^{- 2} d x \\ = {[\frac{(x + 5)^{- 2 + 1}}{- 2 + 1}]}_{- 3}^{2} \\ = {[\frac{(x + 5)^{- 1}}{- 1}]}_{- 3}^{2} \\ = {[\frac{- 1}{(x + 5)}]}_{- 3}^{2} \\ = \frac{- 1}{(2 + 5)} + \frac{1}{(- 3 + 5)} \\ = \frac{- 1}{7} + \frac{1}{2} \\ = \frac{- 2 + 7}{14} \\ = \frac{5}{14} \end{aligned}$

Therefore, the exact value is $\frac{5}{14}$ . $\frac{5}{14}$

Step 3. Check:

The required graph is,

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