Chapter 5: Q. 37 (page 464)

Solve $\int \frac{x}{4 + x^{2}} d x$ 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 $\int \frac{x}{4 + x^{2}} d x .$

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,

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

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,

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

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Solve $\int \frac{x}{4 + x^{2}} d x$ 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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Solve∫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

Statistics

Mechanics Maths

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Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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