/*! 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 9 Find the indefinite integral and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 9

Find the indefinite integral and check the result by differentiation. $$ \int(1+2 x)^{4}(2) d x $$

Short Answer

Expert verified
The indefinite integral of the function is \( \frac{1}{5}(1+2x)^5 + C \).

Step by step solution

01

Perform Integration

The given integral can be written in a more digestible format as: \( \int 2(1+2x)^4 dx \). In order to perform this integration, a substitution method can be used. Let the substitution \( u = 1 + 2x \). This gives \( du = 2dx \). Therefore, the integral becomes: \( \int u^4 du \). Solving this gives the integrated function as \( \frac{1}{5}u^5 + C \), where C is the constant of integration.
02

Substitute Back

Now, substituting u back in terms of x, the solution becomes \( \frac{1}{5}(1+2x)^5 + C \). This is the indefinite integral of the given function.
03

Check the Result by Differentiation

Now, let's ensure correctness by differentiating the obtained function. The rule of differentiation states that the derivative of an indefinite integral of a function is the original function. Hence, differentiate \( \frac{1}{5}(1+2x)^5 \) using the power and chain rule, and this should give the original function, \( 2 (1+2x)^4 \).

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 the value \(\int_{0}^{1} x^{2} d x=\frac{1}{3}\) to evaluate each definite integral. Explain your reasoning. (a) \(\int_{-1}^{0} x^{2} d x\) (b) \(\int_{-1}^{1} x^{2} d x\) (c) \(\int_{0}^{1}-x^{2} d x\)

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{2}^{5}\left(\frac{1}{x^{2}}-\frac{1}{x^{3}}\right) d x $$

State whether the function is even, odd, or neither. $$ f(x)=3 x^{4} $$

Determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g\). (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=2-\frac{1}{2} x, \quad g(x)=2-\sqrt{x}\) (a) 1 (b) 6 (c) \(-3\) (d) 3 (e) 4

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{1 / 2}^{1}(x+1) \sqrt{1-x} d x $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.