Q. 37 Solve鈭玿4+x2dx聽the following t... [FREE SOLUTION]

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Chapter 5: Q. 37 (page 464)

Solve $鈭� \frac{x}{4 + x^{2}} d x$ the following two ways:
(a) with the substitution $u = 4 + x^{2};$
(b) with the trigonometric substitution x = 2 tan u.

Short Answer

Expert verified

Part (a) The solution of the integral is $\frac{1}{2} \ln |4 + x^{2}| + C .$

Part (b) The solution of the integral is $- \ln |\frac{2}{\sqrt{4 + x^{2}}}| + C .$

Step by step solution

Part (a) Step 1. Given Information.

The given integral is $鈭� \frac{x}{4 + x^{2}} d x .$

Part (a) Step 2. Solve.

We have to solve the integral with the substitution $u = 4 + x^{2},$ so the derivative is $d u = (2 x) d x .$

Let's solve the integral by substituting u,

$鈭� \frac{x}{4 + x^{2}} d x = 鈭� \frac{x}{u} (\frac{d u}{2 x}) = \frac{1}{2} 鈭� \frac{d u}{u} = \frac{1}{2} [\ln |u|] + C Substitue back u, = \frac{1}{2} \ln |4 + x^{2}| + C$

Part (b) Step 1. Solve.

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

Let's solve the integral by substituting x,

$鈭� \frac{x}{4 + x^{2}} d x = 鈭� \frac{2 \tan u}{4 + {(2 \tan u)}^{2}} (2 s e c^{2} u) d u = 鈭� \frac{4 \tan u s e c^{2} u}{4 + (4 \tan^{2} u)} d u = 鈭� \frac{4 \tan u s e c^{2} u}{4 (1 + \tan^{2} u)} d u = 鈭� \frac{\tan u s e c^{2} u}{(1 + \tan^{2} u)} d u Use the identity 1 + \tan^{2} x = \sec^{2} x = 鈭� \frac{tanu \sec^{2} u}{(\sec^{2} u)} du = 鈭� tanu du = - \ln |cosu| + C Substitute back u, = - \ln |\cos (\tan^{- 1} (\frac{x}{2}))| + C = - \ln |\frac{2}{\sqrt{4 + x^{2}}}| + C$

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91影视

Solve $鈭� \frac{x}{4 + x^{2}} d x$ the following two ways:
(a) with the substitution $u = 4 + x^{2};$
(b) with the trigonometric substitution x = 2 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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91影视

Solve鈭�x4+x2dxthe following two ways:(a) with the substitution u=4+x2;(b) with the trigonometric substitution x = 2 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

Mechanics Maths

Statistics

Geometry

Applied Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Solve $鈭� \frac{x}{4 + x^{2}} d x$ the following two ways:
(a) with the substitution $u = 4 + x^{2};$
(b) with the trigonometric substitution x = 2 tan u.