/*! 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 17 Find the indefinite integral and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 17

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{2}}{\left(1+x^{3}\right)^{2}} d x $$

Short Answer

Expert verified
The indefinite integral of \( \frac{x^{2}}{(1+x^{3})^{2}} \) is \( -\frac{1}{3(1+x^3)} + C \).

Step by step solution

01

Identifying the substitution

Looking at the integral, it is seen that the numerator is the derivative of the denominator. This makes it a candidate for substitution. Let's let \( u = 1 + x^3 \). The derivative of \( u \) with respect to \( x \) is \( du/dx = 3x^2 \), or \( dx = du / (3x^2) \). Write these substitutions down.
02

Substitute and Simplify

Replace \( dx \) and \( 1 + x^3 \) in the original integral with the substitutions determined in step 1. The integral becomes: \[ \int \frac{x^2}{u^2} \cdot \frac{du}{3x^2} \] The \( x^2 \) in the numerator and denominator cancel out to leave: \[ \int \frac{1}{3u^2} du \] Now the integral can be solved.
03

Solve the simplified integral

The integral is now much simpler and can be solved using basic integral rules: \[ \int \frac{1}{3u^2} du = -\frac{1}{3} \cdot \frac{1}{u} + C = -\frac{1}{3u} + C \] where \( C \) is the constant of integration.
04

Substitute the expression for u

Now substitute back the original expression for \( u \) (which was \( 1+x^3 \)) into the solution from step 3. This gives the following solution: \[ -\frac{1}{3(1+x^3)} + C \] This completes the integration.
05

Verify the result

Finally, to verify the result, differentiate the answer with respect to \( x \). If the derivative of the answer is equal to the original integrand, that confirms the solution: \[ \frac{d}{dx} \left(-\frac{1}{3(1+x^3)} + C\right) = \frac{x^2}{(1+x^3)^2} \] which is indeed the function we started with.

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

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{1}{2} x\right] d x $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=x^{3}-2 x+1, y=-2 x, x=1 $$

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 250, \quad r=8 \%, \quad T=6 \text { years } $$

The projected fuel cost \(C\) (in millions of dollars per year) for an airline company from 2007 through 2013 is \(C_{1}=568.5+7.15 t\), where \(t=7\) corresponds to \(2007 .\) If the company purchases more efficient airplane engines, fuel cost is expected to decrease and to follow the model \(C_{2}=525.6+6.43 t\). How much can the company save with the more efficient engines? Explain your reasoning.

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=3 x^{2}+1 \quad[-1,3] $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.