Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Evaluate the integrals.
\(\int_{ - 10}^{10} {\frac{{2{e^x}}}{{\sinh x + \cosh x}}} dx\)

The value of \(\int_{ - 10}^{10} {\frac{{2{e^x}}}{{\sinh x + \cosh x}}} dx\)is \(40\).

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1 Definition of the integral

Integral calculus is an area of mathematics that deals with the calculation, properties, and applications of integrals.

Step 2 : Evaluating the integral.

The given data is,

\(\int_{ - 10}^{10} {\frac{{2{e^x}}}{{\sinh x + \cosh x}}} dx\)

Given that: \(\sinh x = \frac{{{e^x} - {e^{ - x}}}}{2}\) and \(\cosh x = \frac{{{e^x} + {e^{ - x}}}}{2}\)

Therefore

\(\begin{aligned}{c}\sinh x + \cosh x = \frac{{{e^x} - {e^{ - x}}}}{2} + \frac{{{e^x} + {e^{ - x}}}}{2}\\ = \frac{{2{e^x}}}{2}\\ = {e^x}\\ = \int_{ - 10}^{10} {\frac{{2{e^x}}}{{{e^x}}}} dx\\ = \int_{ - 10}^{10} 2 dx\\ = \int_{ - 10}^{10} 2 \cdot {x^0}dx\end{aligned}\)

Step 3 : Using the power rule of integration, which states that for \(n \ne - 1\)

Then it is expressed as,

\(\begin{aligned}{c}\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C\\ = \left( {2 \times \frac{{{x^{0 + 1}}}}{{0 + 1}}} \right)_{ - 10}^{10}\\ = (2x)_{ - 10}^{10}\\ = (2 \times 10 - 2 \times ( - 10))\\ = 20 + 20\\ = 40\end{aligned}\)

Hence the value of \(\int_{ - 10}^{10} {\frac{{2{e^x}}}{{\sinh x + \cosh x}}} dx\) is 40.

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the integrals.
\(\int_{ - 10}^{10} {\frac{{2{e^x}}}{{\sinh x + \cosh x}}} dx\)

Step by Step Solution

Step 1 Definition of the integral

Step 2 : Evaluating the integral.

Step 3 : Using the power rule of integration, which states that for \(n \ne - 1\)

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the integrals.\(\int_{ - 10}^{10} {\frac{{2{e^x}}}{{\sinh x + \cosh x}}} dx\)

Step by Step Solution

Step 1 Definition of the integral

Step 2 : Evaluating the integral.

Step 3 : Using the power rule of integration, which states that for \(n \ne - 1\)

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Evaluate the integrals.
\(\int_{ - 10}^{10} {\frac{{2{e^x}}}{{\sinh x + \cosh x}}} dx\)