/*! 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 3 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Evaluate the following integrals : $$ \int x^{-1}\left(1+x^{1 / 3}\right)^{-3} d x $$

Short Answer

Expert verified
Question: Evaluate the integral $\int x^{-1}(1+x^{1/3})^{-3}dx$. Answer: $\frac{-3(1+x^{1/3})^{-2}}{2} + C$

Step by step solution

01

Identify the substitution

We observe that the function inside the parenthesis can be a good candidate for substitution, so let: $$ u = 1 + x^{1/3} $$
02

Find the derivative

Now we will find the derivative of this substitution with respect to x: $$ \frac{du}{dx} = \frac{d (1 + x^{1/3})}{dx} = \frac{1}{3}x^{-2/3} $$
03

Rearranging for dx

We need to rearrange the equation from Step 2 to find 'dx': $$ dx = \frac{3}{x^{2/3}} du $$
04

Substitute u and dx into the integral

Now substitute u and dx into the original integral: $$ \int x^{-1}(1+x^{1/3})^{-3}dx = \int (u-1)^{2/3}u^{-3}\frac{3}{(u-1)^{2/3}} du $$
05

Simplify the integral

Now we simplify the integral by canceling out the terms: $$ \int (u-1)^{2/3}u^{-3}\frac{3}{(u-1)^{2/3}} du = 3\int u^{-3} du $$
06

Evaluate the integral

Now we can evaluate the simplified integral using the power rule for integration: $$ 3\int u^{-3} du = 3\left(\frac{u^{-2}}{-2}\right) + C = \frac{-3u^{-2}}{2} + C $$
07

Substitute back for x

Finally, we substitute back the original expression for u: $$ u = 1 + x^{1/3} \Rightarrow \frac{-3u^{-2}}{2} + C = \frac{-3(1+x^{1/3})^{-2}}{2} + C $$ So, the evaluated integral is: $$ \int x^{-1}\left(1+x^{1 / 3}\right)^{-3} d x = \frac{-3(1+x^{1/3})^{-2}}{2} + C $$

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

Evaluate the following integrals: $$ \int \frac{x^{2}+2 x+3}{\sqrt{\left(x^{2}+x+1\right)}} d x $$

Evaluate the following integrals: (i) \(\int \mathrm{e}^{\mathrm{x}} \frac{\left(\mathrm{x}^{2}-3 \mathrm{x}+3\right)}{(\mathrm{x}+2)^{2}} \mathrm{dx}\) (ii) \(\int \frac{\mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}+1\right)}{(\mathrm{x}+1)^{2}} \mathrm{dx}\) (iii) \(\int \mathrm{e}^{x} \frac{(1-x)^{2}}{\left(1+x^{2}\right)^{2}} d x\) (iv) \(\int \frac{x^{2} e^{x}}{(x+2)^{2}} d x\)

\(\int \frac{\sqrt{x^{2}+1}}{x^{4}} \ln \left(1+\frac{1}{x^{2}}\right) d x\)

Evaluate the following integrals : (i) \(\int \mathrm{e}^{\mathrm{x}}(\sin \mathrm{x}-\cos \mathrm{x}) \mathrm{dx}\) (ii) \(\int \mathrm{e}^{\mathrm{x}}(\tan \mathrm{x}-\ln \cos x) \mathrm{dx}\) (iii) \(\int \mathrm{e}^{x} \sec x \cdot(1+\tan x) d x\)

Evaluate the following integrals: (i) \(\int \frac{d x}{\sin x(3+2 \cos x)}\) (ii) \(\int \frac{\mathrm{d} \mathrm{x}}{\sin 2 \mathrm{x}-2 \sin \mathrm{x}}\) (iii) \(\int \frac{\sin \frac{\theta}{2} \tan \frac{\theta}{2} \mathrm{~d} \theta}{\cos \theta}\) (iv) \(\int \frac{d x}{\ln x^{x}\left[(\ln x)^{2}-3 \ln x-10\right]}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

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.