Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Solve $\int \frac{x^{2}}{1 + x^{2}} d x$ the following two ways:
(a) with the substitution $u = \tan^{- 1} x;$
(b) with the trigonometric substitution x = tan u.

Part (a) The solution of the given integral is $x - \tan^{- 1} x + C .$

Part (b) The solution of the given integral is $x - \tan^{- 1} x + C .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part (a) Step 1. Given Information.

The given integral is $\int \frac{x^{2}}{1 + x^{2}} d x .$

Part (a) Step 2. Solve.

We have to solve the integral with the substitution $u = \tan^{- 1} x,$ so the derivative is $d u = \frac{1}{1 + x^{2}} d x .$

Let's solve the integral by substituting u,

$\int \frac{x^{2}}{1 + x^{2}} d x = \int \frac{{(\tan u)}^{2}}{1 + {(\tan u)}^{2}} (1 + {(\tan u)}^{2}) d u = \int \tan^{2} u d u = \int - 1 + s e c^{2} u d u = - \int 1 d u + \int s e c^{2} u d u = - u + \tan u + C Substitute back u = - \tan^{- 1} x + x + C = x - \tan^{- 1} x + C$

Part (b) Step 1. Solve.

We have to solve the integral with the substitution $x = \tan u,$ so the derivative is $d x = s e c^{2} u d u .$

Let's solve the integral by substituting x,

role="math" localid="1648810032401" $\int \frac{x^{2}}{1 + x^{2}} d x = \int \frac{\tan^{2} u}{1 + \tan^{2} u} s e c^{2} u d u = \int \frac{\tan^{2} u}{s e c^{2} u} s e c^{2} u d u [1 + \tan^{2} u = s e c^{2} u] = \int \tan^{2} u d u = \int - 1 + s e c^{2} u d u = - \int 1 d u + \int s e c^{2} u d u = - u + \tan u + C Substitute back u = - \tan^{- 1} x + x + C = x - \tan^{- 1} x + C$

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Calculus

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Short Answer

Solve $\int \frac{x^{2}}{1 + x^{2}} d x$ the following two ways:
(a) with the substitution $u = \tan^{- 1} x;$
(b) with the trigonometric substitution x = tan u.

Step by Step Solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Solve.

Part (b) Step 1. Solve.

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Solve $\int \frac{x^{2}}{1 + x^{2}} d x$ the following two ways:
(a) with the substitution $u = \tan^{- 1} x;$
(b) with the trigonometric substitution x = tan u.