/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Find or evaluate the integral. (... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 23

Find or evaluate the integral. (Complete the square, if necessary.) $$ \int_{0}^{2} \frac{d x}{x^{2}-2 x+2} $$

Short Answer

Expert verified
\(\frac{\pi}{2}\)

Step by step solution

01

Rewrite the function

Rewrite the integral in a simpler form by completing the square in the denominator:\[ \int_{0}^{2} \frac{d x}{x^{2}-2 x+2} = \int_{0}^{2} \frac{dx}{(x-1)^2+1}\]
02

Substitute variables

By using the substitution \(u = x - 1\), find the new limits of integration \(u_{0} = -1\) and \(u_{2} = 1\), and express the integral as follows:\[ \int_{-1}^{1} \frac{du}{u^2+1}\]
03

Evaluate integral

Now evaluate this simpler integral which is a standard form that is memorized / known to be \( \arctan(u) \):\[ \= \left.\arctan(u)\right|_{-1}^{1} = \arctan(1) - \arctan(-1)\]
04

Evaluating Limits

Finally, evaluate the definite integral by substituting the new limits of \(u\) into the antiderivative: \[= \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{2}\]

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!

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Verify the differentiation formula. \(\frac{d}{d x}\left[\cosh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}-1}}\)

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{1}^{x} \frac{t^{2}}{t^{2}+1} d t $$

Show that the function satisfies the differential equation. \(y=a \cosh x\) \(y^{\prime \prime}-y=0\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int \frac{d x}{25+x^{2}}=\frac{1}{25} \arctan \frac{x}{25}+C $$

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x} t \cos t d t $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.