/*! 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 18 Find the indefinite integral by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 18

Find the indefinite integral by \(u\) -substitution. (Hint: Let \(u\) be the denominator of the integrand.) $$ \int \frac{\sqrt[3]{x}}{\sqrt[3]{x}-1} d x $$

Short Answer

Expert verified
The solution to the integral is \( \frac{3}{2} (\sqrt[3]{x}-1)^2 x + x(\sqrt[3]{x}-1) + (\sqrt[3]{x})^3 -1 +C \).

Step by step solution

01

Set up the \(u\)-substitution

The denominator of the integrand is given as \(\sqrt[3]{x}-1\). According to the hint, we should let \(u\) be this value. Therefore, let \(u=\sqrt[3]{x}-1\). This implies that \(x=(u+1)^3\). We differentiate both sides with respect to \(x\) to find the substitution for \(dx\). On differentiating, we get \(dx = 3(u+1)^2 du\). We now substitute \(x\) and \(dx\) in the integral.
02

Substitute \(x\) and \(dx\) in the integral

Substituting the values for \(x\) and \(dx\) in the integral gives: \( \int \frac{(u+1)}{u} \cdot 3(u+1)^2 du\).
03

Simplify the integral

We factor out a 3 from the integral. Then split the fraction and rewrite the integral as two separate integrals. This gives: \(3 \int u(u+1)^2 du + 3 \int (u+1)^2 du\).
04

Evaluate the integrals

Integrating each term separately using the power rule for integration yields the solution. The power rule of integration states that \(\int u^n du = \frac{1}{n+1}u^{n+1}\). On integrating each term, you will get \( \frac{3}{2} u^2 (u+1)^2 + u(u+1)^2 + (u+1)^3 + C \).
05

Substitute back the value of \(u\)

Substitute back \(u = \sqrt[3]{x} - 1\) to get the answer in terms of \(x\). This gives: \( \frac{3}{2} (\sqrt[3]{x}-1)^2 x + x(\sqrt[3]{x}-1) + (\sqrt[3]{x})^3 -1 +C \).

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

Find the indefinite integral using the formulas of Theorem 4.24 \(\int \frac{1}{\sqrt{x} \sqrt{1+x}} d x\)

(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{\pi / 4}^{x} \sec ^{2} t d t $$

In Exercises \(79-84,\) find \(F^{\prime}(x)\). $$ F(x)=\int_{x}^{x+2}(4 t+1) d t $$

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

In Exercises \(27-30,\) find any relative extrema of the function. Use a graphing utility to confirm your result. \(f(x)=\sin x \sinh x-\cos x \cosh x, \quad-4 \leq x \leq 4\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Decision 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.