Chapter 5: Q. 34 (page 464)

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.

Short Answer

Expert verified

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

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

Step by step solution

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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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.

Short Answer

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∫x21+x2dx the following two ways:(a) with the substitution u=tan-1x;(b) with the trigonometric substitution x = tan u.

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Solve.

Part (b) Step 1. Solve.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

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Calculus

Study anywhere. Anytime. Across all devices.

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.