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Expert-verified Found in: Page 373 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals.Use a graph to check your answer. ${\int }_{-3}^{2} \frac{1}{\left(x+5{\right)}^{2}}dx$

Ans: The exact value is, $\phantom{\rule{0ex}{0ex}}{\int }_{-3}^{2} \frac{1}{\left(x+5{\right)}^{2}}dx=\frac{5}{14}$

See the step by step solution

## Step 1. Given information.

given expression,

${\int }_{-3}^{2} \frac{1}{\left(x+5{\right)}^{2}}dx$

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

The exact value is calculated as shown below,

$\begin{array}{r}{\int }_{-3}^{2} \frac{1}{\left(x+5{\right)}^{2}}dx\\ ={\int }_{-3}^{2} \left(x+5{\right)}^{-2}dx\\ ={\left[\frac{\left(x+5{\right)}^{-2+1}}{-2+1}\right]}_{-3}^{2}\\ ={\left[\frac{\left(x+5{\right)}^{-1}}{-1}\right]}_{-3}^{2}\\ ={\left[\frac{-1}{\left(x+5\right)}\right]}_{-3}^{2}\\ =\frac{-1}{\left(2+5\right)}+\frac{1}{\left(-3+5\right)}\\ =\frac{-1}{7}+\frac{1}{2}\\ =\frac{-2+7}{14}\\ =\frac{5}{14}\end{array}$

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

## Step 3. Check:

The required graph is,  ### Want to see more solutions like these? 