Q. 48 Solve each of the integrals in E... [FREE SOLUTION]

Chapter 5: Q. 48 (page 417)

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$鈭� \frac{1 鈭� x}{1 + 3 x^{2}} d x$

Short Answer

Expert verified

The solution of the given integral is $鈭� \frac{1 鈭� x}{1 + 3 x^{2}} d x = \frac{1}{3} \tan^{- 1} 3 x 鈭� \frac{1}{6} \ln (1 + 3 x^{2}) + C$ .

Step by step solution

Step 1. Given Information

Solving the given integrals.

$鈭� \frac{1 鈭� x}{1 + 3 x^{2}} d x$

Step 2. Solving the given integral using substitution method.

Let

$u = 1 + 3 x^{2} v = 3 x \frac{d u}{d x} = 6 x \frac{d v}{d x} = 3 d u = 6 x d x d v = 3 d x \frac{1}{6} d u = x d x \frac{1}{3} d v = d x$

Step 3. This substitution changes the integral into

$鈭� \frac{1 鈭� x}{1 + 3 x^{2}} d x = 鈭� (\frac{1}{1 + 3 x^{2}} 鈭� \frac{x}{1 + 3 x^{2}}) d x 鈭� \frac{1 鈭� x}{1 + 3 x^{2}} d x = 鈭� \frac{1}{1 + 3 x^{2}} d x 鈭� 鈭� \frac{x}{1 + 3 x^{2}} d x 鈭� \frac{1 鈭� x}{1 + 3 x^{2}} d x = \frac{1}{3} 鈭� \frac{1}{1 + v^{2}} dv 鈭� \frac{1}{6} 鈭� \frac{1}{u} du 鈭� \frac{1 鈭� x}{1 + 3 x^{2}} dx = \frac{1}{3} \tan^{- 1} v 鈭� \frac{1}{6} \ln (u) + C 鈭� \frac{1 鈭� x}{1 + 3 x^{2}} dx = \frac{1}{3} \tan^{- 1} 3 x 鈭� \frac{1}{6} \ln (1 + 3 x^{2}) + C$

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91影视

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$鈭� \frac{1 鈭� x}{1 + 3 x^{2}} d x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the given integral using substitution method.

Step 3. This substitution changes the integral into

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91影视

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)鈭�1鈭�x1+3x2dx

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the given integral using substitution method.

Step 3. This substitution changes the integral into

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Mechanics Maths

Pure Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$鈭� \frac{1 鈭� x}{1 + 3 x^{2}} d x$