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Q. 37

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.
$\int_{- 1}^{1} \frac{2^{x}}{4^{x}} d x$

Ans: The exact value of, $\int_{- 1}^{1} \frac{2^{x}}{4^{x}} d x = \frac{3}{2 \ln (2)}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

given,

$\int_{- 1}^{1} \frac{2^{x}}{4^{x}} d x$

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

The exact value is calculated as shown below,

$\begin{aligned} \int_{- 1}^{1} \frac{2^{x}}{4^{x}} d x \\ = \int_{- 1}^{1} 2^{x} 4^{- x} d x \\ = \int_{- 1}^{1} 2^{x} 2^{- 2 x} d x \\ = \int_{- 1}^{1} 2^{- x} d x \\ = {[- \frac{1}{\ln (2)} 2^{- x}]}_{- 1}^{1} \\ = {[- \frac{1}{2^{x} \ln (2)}]}_{- 1}^{1} \\ = [- \frac{1}{2^{1} \ln (2)} + \frac{1}{2^{- 1} \ln (2)}] \\ = \frac{3}{2 \ln (2)} \end{aligned}$

Therefore, the value is $\frac{3}{2 \ln (2)}$ .

Step 3. Check

The required graph is,

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