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Chapter 4: Problem 474

Find the antiderivative \(F(x)\) of each function \(f(x)\). \(\quad f(x)=5 x^{4}+4 x^{5}\)

Short Answer

Expert verified
The antiderivative is \( F(x) = x^5 + \frac{2x^6}{3} + C \).

Step by step solution

01

Understand the Problem

The problem asks for the antiderivative, or indefinite integral, of the function \( f(x) = 5x^4 + 4x^5 \). This means we need to find a function \( F(x) \) such that \( F'(x) = f(x) \).
02

Apply the Power Rule for Integration

Recall that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), assuming \( n eq -1 \). We will apply this rule to each term in \( f(x) \).
03

Integrate the First Term

Integrate \( 5x^4 \):\[ \int 5x^4 \, dx = 5 \cdot \frac{x^{4+1}}{4+1} = \frac{5x^5}{5} = x^5 \].
04

Integrate the Second Term

Integrate \( 4x^5 \):\[ \int 4x^5 \, dx = 4 \cdot \frac{x^{5+1}}{5+1} = \frac{4x^6}{6} = \frac{2x^6}{3} \].
05

Combine the Results and Add Constant of Integration

Combine the integrals of the terms and add the constant of integration \( C \):\[ F(x) = x^5 + \frac{2x^6}{3} + C \].
06

Final Review

Ensure all calculations are correct and that the antiderivative found is indeed the function whose derivative equals \( f(x) \).

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Key Concepts

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

Power Rule
The power rule is a straightforward and essential tool for finding the antiderivative, or indefinite integral, of algebraic functions. It states that for any term of the form \(x^n\), its antiderivative is \( \frac{x^{n+1}}{n+1} \) as long as \(n eq -1\).
This rule allows us to reverse the process of differentiation easily, making it a key step in calculus for both differentiation and integration processes.
Let's see how the power rule transforms the terms in our example function:
  • For \(5x^4\), we apply the power rule: \( \int 5x^4 \, dx = \frac{5x^{5}}{5} = x^5\).
  • For \(4x^5\), we apply the power rule: \( \int 4x^5 \, dx = \frac{4x^{6}}{6} = \frac{2x^6}{3}\).
These steps assign each term a corresponding antiderivative term without much complexity, making the power rule an elementary and efficient tool in calculus.
Indefinite Integral
In calculus, the indefinite integral represents a family of functions whose derivatives result in the given function. Unlike a definite integral, which calculates the area under a curve within specific limits, an indefinite integral provides a general solution encompassing all possible antiderivatives.
It is expressed using the integral sign \(\int\), and for a function \(f(x)\), the indefinite integral is \( \int f(x) \, dx \). In our problem:
  • We need to calculate \( \int (5x^4 + 4x^5) \, dx \).
  • By integrating each term separately, we find the antiderivative function as \( F(x) = x^5 + \frac{2x^6}{3}\).
Therefore, the indefinite integral offers a powerful method to reverse differentiation, leading to the original function, plus a constant.
Integration Constant
The integration constant, often denoted as \( C \), is an essential component when finding indefinite integrals.
When we integrate a function, we consider all possible vertical shifts, since differentiation erases any constant added to a function. This is why the integration constant \( C \) is included, representing unknown numbers added or subtracted from our integral function.
In our example, after finding the antiderivative of each term:
  • \( F(x) = x^5 + \frac{2x^6}{3} + C \).
  • The \( C \) accounts for any constant that could have been in the original function before differentiation.
This small yet crucial detail ensures that our solutions cover all possible original functions, restoring the completeness lost in differentiation.

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