/*! 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 32 Determine the following indefini... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 32

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z$$

Short Answer

Expert verified
Question: Find the indefinite integral of the given function and verify the solution by differentiation: $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z$$ Answer: The indefinite integral of the given function is $$3z^{4/3} -\frac{3}{2}z^{2/3} + C$$

Step by step solution

01

Separate the Function into Two Integrals

We have: $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z = \int 4z^{1/3}dz - \int z^{-1/3} dz$$
02

Integrate Each Function

Integrate each side separately: $$\int 4z^{1/3}dz = 4\int z^{1/3}dz = \frac{4}{4/3} z^{(1/3)+1} + C_1 = 3z^{4/3} + C_1$$ $$\int z^{-1/3} dz = \frac{1}{(-1/3)+1} z^{(-1/3)+1} + C_2 = -\frac{3}{2} z^{2/3} + C_2$$ Now, we combine the two integrals: $$\int\left(4 z^{1/3}-z^{-1/3}\right) d z = 3z^{4/3} - \frac{3}{2} z^{2/3} + C = G(z)$$ Where C is the constant of integration (C = C_1 - C_2).
03

Check by Differentiation

Take the derivative of the integrated function: $$\frac{d}{dz} G(z) = \frac{d}{dz} \left( 3z^{4/3} - \frac{3}{2} z^{2/3} + C \right)$$ Use the power rule for differentiation to find: $$\frac{d}{dz} G(z) = \left(4z^{(4/3)-1}\right) - \left(\frac{3}{2}(2/3)z^{(2/3)-1}\right) = 4z^{1/3} - z^{-1/3}$$ This is the original function, so our integration and solution are correct. The final answer is: $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z = 3z^{4/3} - \frac{3}{2} z^{2/3} + C$$

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 analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty}(\sqrt{x-2}-\sqrt{x-4})$$

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=2 \cos t ; v(0)=1, s(0)=0$$

Differentials Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=\sin ^{-1} x$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The linear approximation to \(f(x)=x^{2}\) at \(x=0\) is \(L(x)=0\) b. Linear approximation at \(x=0\) provides a good approximation to \(f(x)=|x|\) c. If \(f(x)=m x+b,\) then the linear approximation to \(f\) at any point is \(L(x)=f(x)\)

Use analytical methods to evaluate the following limits. $$\lim _{t \rightarrow \pi / 2^{+}} \frac{\tan 3 t}{\sec 5 t}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

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.