Chapter 5: Q. 45 (page 417)

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$\int \frac{\ln \sqrt{x}}{x} d x$

Short Answer

Expert verified

The solution of the given integral is $\int \frac{\ln \sqrt{x}}{x} d x = \frac{{[\ln \sqrt{x}]}^{2}}{4} + C$ .

Step by step solution

Step 1. Given Information

Solving the given integrals.

$\int \frac{\ln \sqrt{x}}{x} d x$

Step 2. Solving the given integral using substitution method.

Let

$u = \ln \sqrt{x} \frac{d u}{d x} = \frac{1}{\sqrt{x}} \cdot \frac{1}{2 \sqrt{x}} \frac{d u}{d x} = \frac{1}{2 x} \frac{1}{2} d u = \frac{1}{x} d x$

Step 3. This substitution changes the integral into

$\int \frac{\ln \sqrt{x}}{x} d x = \frac{1}{2} \int u d u \int \frac{\ln \sqrt{x}}{x} d x = \frac{1}{2} [\frac{u^{1 + 1}}{1 + 1}] + C \int \frac{\ln \sqrt{x}}{x} d x = \frac{1}{2} [\frac{u^{2}}{2}] + C \int \frac{\ln \sqrt{x}}{x} d x = \frac{u^{2}}{4} + C \int \frac{\ln \sqrt{x}}{x} d x = \frac{{[\ln \sqrt{x}]}^{2}}{4} + C$

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Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$\int \frac{\ln \sqrt{x}}{x} d x$

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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Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)∫lnxxdx

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

Decision Maths

Geometry

Probability and Statistics

Pure Maths

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$\int \frac{\ln \sqrt{x}}{x} d x$