/*! 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 44 Find or evaluate the integral us... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 44

Find or evaluate the integral using substitution first, then using integration by parts. $$ \int \ln \left(x^{2}+1\right) d x $$

Short Answer

Expert verified
The integral evaluates to \( \frac{1}{2} (x^2 + 1) \ln(x^2 + 1) - \frac{1}{2} (x^2 + 1) + C \)

Step by step solution

01

Set up for substitution

First, we use the substitution \( u = x^2 + 1 \). The derivative \( du = 2x dx \). We can rewrite the intergral as follows \( \frac{1}{2} \int \ln(u) du \). Now the integral looks simpler.
02

Set up for integration by parts

Now we will use the integration by parts on the simpler representation of the integral. By using integration by parts, the formula is \(\int u v dx = u v - \int v du\). Let \( v = \ln(u) \) and \( dv = \frac{1}{u} du \). Then \( u = u \) and \( du = du \). After computing these, it is easier to compute the integral.
03

Apply integration by parts

Apply the formula \(\int u v dx = u v - \int v du\) as follows: \( \frac{1}{2} = \frac{1}{2} u \ln(u) - \frac{1}{2} \int du = \frac{1}{2} u \ln(u) - \frac{1}{2} u + C\). This is the result of the first method. Notice that we must convert \(u\) to \(x\).
04

Convert back to x

Now that we have our result in terms of \(u\), we need to change everything back in terms of \(x\). Substitute back \(u = x^2 + 1\) from Step 1 to obtain the final answer: \( \frac{1}{2} (x^2 + 1) \ln(x^2 + 1) - \frac{1}{2} (x^2 + 1) + C \).

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.

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

Use a computer algebra system to evaluate the definite integral. In your own words, describe how you would integrate \(\int \sec ^{m} x \tan ^{n} x d x\) for each condition. (a) \(m\) is positive and even. (b) \(n\) is positive and odd. (c) \(n\) is positive and even, and there are no secant factors. (d) \(m\) is positive and odd, and there are no tangent factors.

Determine all values of \(p\) for which the improper integral converges. $$ \int_{1}^{\infty} \frac{1}{x^{p}} d x $$

In Exercises 65 and 66, apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) \(\begin{array}{l} \underline{\text { Functions }} \\ f(x)=\ln x, \quad g(x)=x^{3} \end{array} \quad \frac{\text { Interval }}{\left[1,4\right]}\)

Consider the limit \(\lim _{x \rightarrow 0^{+}}(-x \ln x)\) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. (c) Use a graphing utility to verify the result of part (b). FOR FURTHER INFORMATION For a geometric approach to this exercise, see the article "A Geometric Proof of \(\lim _{l \rightarrow 0^{+}}(-d \ln d)=0\) " by John H. Mathews in the College Mathematics Journal. To view this article, go to the website www.matharticles.com.

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{\cot ^{2} t}{\csc t} d t $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.