/*! 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 44 Find the indefinite integrals. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 44

Find the indefinite integrals. $$ \int\left(x^{2}+4 x-5\right) d x $$

Short Answer

Expert verified
The indefinite integral is \(\frac{x^3}{3} + 2x^2 - 5x + C\).

Step by step solution

01

Understand the Problem

The task is to find the indefinite integral of the polynomial function \(x^2 + 4x - 5\). This requires finding the antiderivative of each term separately.
02

Apply the Power Rule to Each Term

The power rule for integration states that \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \(n eq -1\). Apply this rule to each term of the polynomial.
03

Integrate the First Term

For the first term \(x^2\), the integral is \(\frac{x^{2+1}}{2+1} = \frac{x^3}{3}\).
04

Integrate the Second Term

For the second term \(4x\), the integral is \(4 \cdot \frac{x^{1+1}}{1+1} = 4 \cdot \frac{x^2}{2} = 2x^2\).
05

Integrate the Third Term

For the third term \(-5\), which is a constant, the integral is \(-5x\), since \( \int a \cdot dx = ax + C\), where \(a\) is a constant.
06

Combine the Integrals

Combine all the results to get the final expression: \(\frac{x^3}{3} + 2x^2 - 5x + C\), where \(C\) is the constant of integration.

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.

Polynomial Integration
Integration is a process where we find the antiderivative or the indefinite integral of a function. When dealing with polynomials, this process involves integrating each term separately.
Polynomials are expressions consisting of variables raised to whole number powers, like the expression in our original exercise:
  • \(x^2\)
  • \(4x\)
  • -5
Finding the indefinite integral of a polynomial means we need to find the integral of each term independently. When solving, make sure to treat each variable term according to its power and treat constant terms using the rules of constant integration. This step-by-step handling ensures accuracy and helps in understanding how each part of the polynomial contributes to the integral.
Power Rule for Integration
The power rule for integration is a key tool when integrating polynomial functions. It provides a formula that simplifies finding integrals of terms with powers.
The power rule states:
  • \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
  • Where \(n eq -1\)
Here's how it works:- Take a term like \(x^2\). According to the power rule, the integral is computed by adding 1 to the exponent \(n\), resulting in \(x^{3}\). Then, divide by this new exponent, \(\frac{x^{3}}{3}\).- For terms like \(4x\), where \(n=1\), add 1 to get \(x^2\) and then divide by 2, multiply by the constant 4, to get \(2x^2\).This method ensures each powered term is integrated correctly and helps in easily identifying the structure of the antiderivative.
Constant of Integration
The constant of integration, represented as \(C\), is a crucial aspect of finding indefinite integrals. In the world of antiderivatives, any constant added to a function鈥檚 derivative won鈥檛 change the derivative itself.
That's why indefinite integrals always include "+ C."- This constant accounts for any fixed value potentially present in the original function before differentiation that would otherwise be "lost."- When given a solution like \(\frac{x^3}{3} + 2x^2 - 5x + C\), the \(C\) ensures completeness.Remember:
  • Every indefinite integral has a family of functions it represents, all differing by some constant \(C\).
  • The concept of \(C\) is vital because it acknowledges that not all functions start from the same point on a graph.
Understanding this idea is key to grasping why indefinite integrals are fundamental in calculus.

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

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{3} e^{0.05 t} d t $$

Estimate \(\int_{0}^{10} f(x) g^{\prime}(x) d x\) if \(f(x)=x^{2}\) and \(g\) has the values in the following table. $$ \begin{array}{c|c|c|c|c|c|c} \hline x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline g(x) & 2.3 & 3.1 & 4.1 & 5.5 & 5.9 & 6.1 \\ \hline \end{array} $$

A firm's marginal cost function is \(M C=3 q^{2}+4 q+6\). Find the total cost function if the fixed costs are 200 .

Graph \(y=1 / x^{2}\) and \(y=1 / x^{3}\) on the same axes. Which do you think is larger: \(\int_{1}^{\infty} 1 / x^{2} d x\) or \(\int_{1}^{\infty} 1 / x^{3} d x\) ? Why?

During a surge in the demand for electricity, the rate, \(r\), at which energy is used can be approximated by $$ r=t e^{-a t} $$ where \(t\) is the time in hours and \(a\) is a positive constant. (a) Find the total energy, \(E\), used in the first \(T\) hours. Give your answer as a function of \(a .\) (b) What happens to \(E\) as \(T \rightarrow \infty\) ?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

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.