Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Evaluate the Integral: \(\int {{{\tan }^2}} \theta {\sec ^4}\theta d\theta \)

To evaluate the Integral\(\int {{{\tan }^2}} \theta {\sec ^4}\theta d\theta \)we will use identity \({\sec ^2}\theta = 1 + {\tan ^2}\theta \) then use the substitution method to get the result.

\(\int {{{\tan }^2}} \theta {\sec ^4}\theta d\theta = \frac{1}{3}{\tan ^3}\theta + \frac{1}{5}{\tan ^5}\theta + c\)

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Splitting the fourth power of the secant function

Let I = \(\int {{{\tan }^2}} \theta {\sec ^4}\theta d\theta \)

First, we will write \({\sec ^4}\theta = {\sec ^2}\theta {\sec ^2}\theta \) then use \({\sec ^2}\theta = 1 + {\tan ^2}\theta \)

Step 2: Using trigonometric identity

\(\begin{aligned}{l}I &= \int {{{\tan }^2}} \theta {\sec ^2}\theta {\sec ^2}\theta d\theta \\ &= \int {{{\tan }^2}} \theta {\sec ^2}\theta (1 + {\tan ^2}\theta )d\theta \end{aligned}\)

Step 3: Substitution method

Now we will use the substitution method

Put \(\tan \theta = x( = ){\sec ^2}\theta d\theta = dx\)

\(I = \int {({x^2}} + {x^4})dx\)

Split the integral

\(\begin{aligned}{l}I = \int {{x^2}} dx + \int {{x^4}} dx\\I = \frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} + c\end{aligned}\)

Step 4 : Re-Substitute the value of \(x = \tan \theta \)

\(I = \frac{{{{\tan }^3}\theta }}{3} + \frac{{{{\tan }^5}\theta }}{5} + c\)

Hence, the value of integral:

\(\int {{{\tan }^2}} \theta {\sec ^4}\theta d\theta = \frac{1}{3}{\tan ^3}\theta + \frac{1}{5}{\tan ^5}\theta + c\)

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the Integral: \(\int {{{\tan }^2}} \theta {\sec ^4}\theta d\theta \)

Step by Step Solution

Step 1: Splitting the fourth power of the secant function

Step 2: Using trigonometric identity

Step 3: Substitution method

Step 4 : Re-Substitute the value of \(x = \tan \theta \)

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the Integral: \(\int {{{\tan }^2}} \theta {\sec ^4}\theta d\theta \)

Step by Step Solution

Step 1: Splitting the fourth power of the secant function

Step 2: Using trigonometric identity

Step 3: Substitution method

Step 4 : Re-Substitute the value of \(x = \tan \theta \)

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