Problem 29 Evaluate the integral. $$\int_... [FREE SOLUTION]

Chapter 16: Problem 29

Evaluate the integral. $$\int_{1}^{\sqrt{3}} \frac{\sqrt{\arctan x}}{1+x^{2}} d x$$

Short Answer

Expert verified

The value of the integral is $ \frac{2}{3} \left( \left( \frac{\pi}{3} \right)^{3/2} - \left( \frac{\pi}{4} \right)^{3/2} \right) $.

Step by step solution

Choose a substitution

Let's start by choosing a substitution to simplify the integral. A good choice here is to let $ u = \arctan x $. This means $ du = \frac{1}{1+x^2} \, dx $, which directly simplifies part of our integrand.

Change limits of integration

Using the substitution $ u = \arctan x $, we need to change the limits of integration. When $ x = 1 $, $ u = \arctan(1) = \frac{\pi}{4} $. When $ x = \sqrt{3} $, $ u = \arctan(\sqrt{3}) = \frac{\pi}{3} $. So the new limits are from $ \frac{\pi}{4} $ to $ \frac{\pi}{3} $.

Rewrite the integral in terms of $ u $

Rewrite the integral in terms of $ u $:\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \sqrt{u} \, du \]This is a much simpler integral to solve.

Integrate with respect to $ u $

Perform the integration. The integral of $ \sqrt{u} $ is:\[ \int \sqrt{u} \, du = \int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C \]Evaluate this from $ u = \frac{\pi}{4} $ to $ u = \frac{\pi}{3} $.

Evaluate the definite integral

Substitute the limits into the integrated function:\[ \frac{2}{3} \left( \left( \frac{\pi}{3} \right)^{3/2} - \left( \frac{\pi}{4} \right)^{3/2} \right) \]Calculate the expression.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integrals

Definite integrals enable us to calculate the area under a curve between two points on the x-axis. This is different from indefinite integrals, which represent a family of functions. A definite integral has limits of integration, expressed as the lower and upper bounds in the integral notation. For example, using the integral $ \int_{1}^{\sqrt{3}} \frac{\sqrt{\arctan x}}{1+x^{2}} \, dx $, the limits of integration are 1 to $ \sqrt{3} $. When handling definite integrals, follow these key steps:

Set the problem up with clear limits of integration.
Apply the Fundamental Theorem of Calculus, which allows evaluation of the integral at these bounds.
Make sure to apply any necessary substitution to simplify the expression before evaluating.

Overall, definite integration is a powerful tool for quantifying areas and other quantities that accumulate over a specific interval.

Integration by Substitution

Integration by substitution is a method used to simplify integrals. It is similar to the chain rule in derivatives. By substituting part of the integrand with a new variable $ u $, the integral can become easier to solve. In our exercise, we made the substitution $ u = \arctan x $. This transformed the integral into a simpler form.Here鈥檚 how to perform a substitution:

Identify a part of the integral function to substitute, naming it $ u $.
Differentiate $ u $ with respect to $ x $, giving $ du $.
Transform the limits of integration, if it is a definite integral, to new limits in terms of $ u $.

For example, the original integral limits $ x = 1 $ and $ x = \sqrt{3} $ change to $ u = \frac{\pi}{4} $ and $ u = \frac{\pi}{3} $, respectively.
Substitution is a vital technique in integral calculus that simplifies problems, making them more manageable.

Mathematical Problem Solving

Mathematical problem solving involves a systematic approach to finding solutions. It's about understanding the problem, planning a strategy, carrying out that plan, and reviewing the solution process. When solving integrals, especially those needing substitution or specific methods like parts, a structured approach is crucial. The following steps help in solving an integral problem systematically:

Understand the function and identify possible simplifications or techniques like substitution.
Clearly define your substitution, ensuring that it represents a part of the integrand that can simplify the process.
Adjust limits of integration and ensure that the bounds correspond with your substitution.
Complete the integration, using the basic integral forms or strengthened techniques for specific functions.
Evaluate the results, substituting back or evaluating specific limits when needed.

Approaching problems with these structured steps ensures a thorough understanding and correct solution with minimized errors.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Evaluate the integral. $$\int_{1}^{\sqrt{3}} \frac{\sqrt{\arctan x}}{1+x^{2}} d x$$

Short Answer

Step by step solution

Choose a substitution

Change limits of integration

Rewrite the integral in terms of \( u \)

Integrate with respect to \( u \)

Evaluate the definite integral

Key Concepts

Definite Integrals

Integration by Substitution

Mathematical Problem Solving

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Mechanics Maths

Applied Mathematics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.