Chapter 5: Q. 34 (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.)
$\int \frac{\sqrt{x} + 1}{x} d x$

Short Answer

Expert verified

The solution of the given integral is $\int \frac{\sqrt{x} + 1}{x} d x = 2 \sqrt{x} + \ln x + C$ .

Step by step solution

Step 1. Given Information

Solving the given integrals.

$\int \frac{\sqrt{x} + 1}{x} d x$

Step 2. Solving the given integral using algebra.

$\int \frac{\sqrt{x} + 1}{x} d x = \int (\frac{\sqrt{x}}{x} + \frac{1}{x}) d x \int \frac{\sqrt{x} + 1}{x} d x = \int (\frac{1}{\sqrt{x}} + \frac{1}{x}) d x \int \frac{\sqrt{x} + 1}{x} d x = \int (\frac{1}{x^{1 / 2}} + \frac{1}{x}) d x \int \frac{\sqrt{x} + 1}{x} d x = \int (x^{- 1 / 2} + \frac{1}{x}) d x$

Step 3. After simplifying

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$\int \frac{\sqrt{x} + 1}{x} d x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the given integral using algebra.

Step 3. After simplifying

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Probability and 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.)∫x+1xdx

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the given integral using algebra.

Step 3. After simplifying

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Probability and 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.)
$\int \frac{\sqrt{x} + 1}{x} d x$