/*! 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 28 Find each indefinite integral. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 28

Find each indefinite integral. $$ \int x^{2}(8 x+3) d x $$

Short Answer

Expert verified
\( \int x^2(8x + 3) \, dx = 2x^4 + x^3 + C \)

Step by step solution

01

Expand the Integrand

The first step is to simplify the integrand by expanding the expression inside the integral. We start with \( x^2(8x + 3) \), which can be expanded as:\[x^2(8x + 3) = x^2 \cdot 8x + x^2 \cdot 3 = 8x^3 + 3x^2\]Now, the integrand becomes \( 8x^3 + 3x^2 \).
02

Integrate Term by Term

The indefinite integral of a sum of functions is the sum of the integrals of each function. So, integrate each term separately:\[ \int (8x^3 + 3x^2) \, dx = \int 8x^3 \, dx + \int 3x^2 \, dx \]Now, find the integral of each term.
03

Apply the Power Rule for Integration

Use the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration:- For \( \int 8x^3 \, dx \): \[ \int 8x^3 \, dx = 8 \cdot \frac{x^{4}}{4} = 2x^4 \]- For \( \int 3x^2 \, dx \): \[ \int 3x^2 \, dx = 3 \cdot \frac{x^{3}}{3} = x^3 \]
04

Combine the Results with a Constant of Integration

Sum the results obtained from the previous step and include the constant of integration, \( C \):\[\int x^2(8x + 3) \, dx = 2x^4 + x^3 + C\]This is the indefinite integral of the given function.

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 technique in calculus. It's used to find the indefinite integral of functions that are powers of a variable.
This rule states: if you want to integrate a function of the form \( x^n \), you use the formula:
  • \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
Here's what it means:
  • Increase the exponent \( n \) by 1 to become \( n+1 \).
  • Divide the result by the new exponent \( n+1 \).
  • Add a constant, \( C \), that accounts for any constant values that could've been differentiated away.
Consider the function \( 8x^3 \). Applying the rule, we increase the exponent from 3 to 4, divide by 4, getting \( 2x^4 \). Similarly, for \( 3x^2 \), it becomes \( x^3 \) after applying the power rule.
Expanding Expressions
Expanding expressions is a crucial step in solving integrals, especially when dealing with polynomials.
For an expression like \( x^2(8x + 3) \), the first task is to distribute terms:
  • Multiply \( x^2 \) by \( 8x \), resulting in \( 8x^3 \).
  • Multiply \( x^2 \) by 3, giving \( 3x^2 \).
This breakdown changes the integrand from a single complex term into simpler parts that can be more easily integrated term by term.
Keeping expressions expanded minimizes mistakes and provides a clearer path forward.
Integration Techniques
Integration involves various methods to find the integral of a function, and in this context, it mainly involves simplifying our work through manageable steps.
Here's how the process often unfolds:
  • **Understand the Structure**: Before applying any mathematical operations, it's important to understand the problem鈥檚 layout and what each part represents.
  • **Simplify the Problem**: As with expanding the expression from \( x^2(8x + 3) \) to \( 8x^3 + 3x^2 \), simplifying makes it easier to apply basic rules.
  • **Integrate Step-by-Step**: Break the problem into parts and tackle each separately. Use known techniques like the power rule for straightforward integration.
Each of these steps builds upon the previous, and when used effectively, they turn a seemingly complex problem into a series of simpler tasks.

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

Suppose that a company found its sales rate (in sales per day) if it did advertise, and also its (lower) sales rate if it did not advertise. If you integrated "upper minus lower" over a month, describe the meaning of the number that you would find.

Sketch each parabola and line on the same graph and find the area between them from \(x=0\) to \(x=3 .\) $$ y=3 x^{2}-12 \text { and } y=2 x-11 $$

Find the average value of each function over the given interval. $$ \begin{array}{l} f(x)=3 \text { on }[10,50]\\\ \text { 14. } f(x)=2 \text { on }[5,100] \end{array} $$

\mathbf{4 5}-\mathbf{6 0} \text { . Find the area bounded by the given curves. } y=x^{2}-1 \text { and } y=2-2 x^{2}

Find the area bounded by the given curves. $$ y=x^{2} \text { and } y=x^{3} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

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.