/*! 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 77 Use the indicated substitution t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 77

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{\sqrt[4]{x+2}+1} ; x+2=u^{4}$$

Short Answer

Expert verified
Question: Find the indefinite integral: $$\int{\frac{dx}{\sqrt[4]{x+2}+1}},$$ given that \(x+2=u^4\). Answer: $$F(x) = -4\left[\frac{1}{3}(\sqrt[4]{x+2})^3- (\sqrt[4]{x+2})^2+ 3\sqrt[4]{x+2}\right]-2(\sqrt[4]{x+2})^2+c$$

Step by step solution

01

Perform the substitution and change the integral bounds

Since it's given that \(x+2=u^4\), we can write \(x=u^4-2\). Now, we can compute the derivative of \(x\) with respect to \(u\). By taking the derivative, we get \(\frac{dx}{du}=4u^3\). So, \({\frac{d x}{d u}} d u={\frac{d x}{d u}}(du)={dx}\). Now, substitute the values of \(x\) and \(dx\) in the given integral: $$\int{\frac{dx}{\sqrt[4]{x+2}+1}}= \int{\frac{4u^3 du}{\sqrt[4]{u^4}+1}}$$
02

Simplify the integral and find the integral of the rational function.

The integral can be simplified as follows: $$\int{\frac{4u^3 du}{\sqrt[4]{u^4}+1}}= \int{\frac{4u^3 du}{u+1}}$$ Now we have a rational function, and we can integrate it using partial fractions: $$\int\frac{4u^3}{u+1}du=-4\int u^2-2u+3-2du$$ Now, we can integrate term by term: $$\: =-4[u^3/3- u^2 + 3u]-2(u^2/2+c)$$
03

Perform the inverse substitution and find the final answer

The final step is to change back the variable to \(x\). Recall that \(u^{4}=x+2,\) so \(u=\sqrt[4]{x+2}\). Substituting back, we get: $$ -4\left[\frac{1}{3}(\sqrt[4]{x+2})^3- (\sqrt[4]{x+2})^2+ 3\sqrt[4]{x+2}\right]-2(\sqrt[4]{x+2})^2+c$$ The final answer to the given integral is: $$F(x) = -4\left[\frac{1}{3}(\sqrt[4]{x+2})^3- (\sqrt[4]{x+2})^2+ 3\sqrt[4]{x+2}\right]-2(\sqrt[4]{x+2})^2+c$$

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 integration by parts to evaluate the following integrals. $$\int_{0}^{1} x \ln x d x$$

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d t}{2+e^{-t}}$$

Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. $$\int \frac{2}{x\left(x^{2}+1\right)^{2}} d x$$

Find the volume of the following solids. The region bounded by \(y=1 /(x+2), y=0, x=0,\) and \(x=3\) is revolved about the line \(x=-1\)

Recall that the substitution \(x=a \sec \theta\) implies that \(x \geq a\) (in which case \(0 \leq \theta<\pi / 2\) and \(\tan \theta \geq 0\) ) or \(x \leq-a\) (in which case \(\pi / 2<\theta \leq \pi\) and \(\tan \theta \leq 0\) ). $$ \begin{array}{l} \text { Show that } \int \frac{d x}{x \sqrt{x^{2}-1}}= \\ \qquad\left\\{\begin{array}{ll} \sec ^{-1} x+C=\tan ^{-1} \sqrt{x^{2}-1}+C & \text { if } x>1 \\ -\sec ^{-1} x+C=-\tan ^{-1} \sqrt{x^{2}-1}+C & \text { if } x<-1 \end{array}\right. \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.