/*! 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 24 Determine these indefinite integ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

Determine these indefinite integrals. $$\int \frac{5}{\sqrt[4]{x^{3}}} d x$$

Short Answer

Expert verified
20x^{\frac{1}{4}} + C

Step by step solution

01

Simplify the integrand

Rewrite the integrand in a simpler form. Recall that \(\frac{5}{\root[4]{x^{3}}} = 5 \times x^{-\frac{3}{4}}\). So the integral becomes: \[ \int \frac{5}{\root[4]{x^{3}}} dx = \int 5 \times x^{-\frac{3}{4}} dx. \]
02

Apply the power rule for integration

The power rule for integration states that \(\frac{x^n}{n+1}\) where \ eq -1\. Applying this rule to our integral: \[ \int 5x^{-\frac{3}{4}} dx = 5 \times \frac{x^{-\frac{3}{4} + 1}}{-\frac{3}{4} + 1} + C. \]
03

Simplify the result

Simplify the exponent and the fraction in the denominator: \[ 5 \times \frac{x^{\frac{1}{4}}}{\frac{1}{4}} + C = 5 \times 4x^{\frac{1}{4}} + C. \]
04

Final answer

Combine the constant coefficients: \[ 20x^{\frac{1}{4}} + C. \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule for Integration
The power rule for integration is a fundamental tool for solving indefinite integrals. This rule states that when integrating a function of the form \( x^n \), we use the formula: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \text{ where } n eq -1. \] Here, 'C' represents the integration constant, which we'll cover later. To apply this rule, simply increase the exponent by 1, then divide by the new exponent. For example: \[ \int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C. \] This method makes integrating polynomials simple and straightforward.
Simplifying Exponents
Before using the power rule for integration, it鈥檚 often necessary to simplify exponents. This means converting complex expressions into a more manageable form. Let's consider the original problem \( \int \frac{5}{\sqrt[4]{x^{3}}} \, dx \). We need to rewrite the integrand so it uses basic power functions. Recall that roots can be expressed as fractional exponents: \[ \frac{1}{\sqrt[4]{x^3}} = x^{-\frac{3}{4}}. \] Now the integral is: \[ \int 5 \times x^{-\frac{3}{4}} \, dx \] Once the exponent is simplified, it's easier to apply the power rule.
Integration Constants
Whenever you compute an indefinite integral, you'll always add a constant of integration, denoted as 'C'. This is because indefinite integrals represent a family of functions, each differing by a constant. In the final step of our problem, we derived: \[ 20x^{\frac{1}{4}} + C. \] The 'C' signifies that any function \( f(x) = 20x^{\frac{1}{4}} + C \) is a valid antiderivative of the integrand we started with. This inclusion ensures we account for all possible vertical shifts of the curve. Always remember to include 'C' in your final answer to represent the complete set of antiderivative functions.

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. $$\int_{1}^{e}\left(x-\frac{1}{x}\right) d x$$

Evaluate. Assume \(u>0\) when ln u appears. $$\int \frac{t^{3} \ln \left(t^{4}+8\right)}{t^{4}+8} d t$$

Approximate the area under the graph of \(F(x)=0.2 x^{3}+2 x^{2}-0.2 x-2\) over the interval [-8,-3] using 5 sub-intervals.

Evaluate. Assume \(u>0\) when ln u appears. $$\int \frac{d x}{a x+b}$$

Industrial learning curve. A company is producing a new product. Due to the nature of the product, the time required to produce each unit decreases as workers become more \intamiliar with the production procedure. It is determined that the function for the learning process is $$T(x)=2+0.3\left(\frac{1}{x}\right)$$ where \(T(x)\) is the time, in hours, required to produce the \(x\) th unit. Use this information. Find the total time required for a new worker to produce units 1 through \(10 ;\) units 20 through 30
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.