Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{2}\left(x^{3}+5 x^{2}-3 x+\sqrt{3}\right) $$

Step-by-Step Solution

Answer

The general antiderivative is \(F(x) = \frac{x^6}{6} + x^5 - \frac{3x^4}{4} + \frac{\sqrt{3}}{3}x^3 + C\).

Step 1: Expand the Function

First, we need to expand the function given. We have \[ f(x) = x^2 (x^3 + 5x^2 - 3x + \sqrt{3}) \] Distribute \(x^2\) to each term inside the parentheses:\[ = x^2 \cdot x^3 + x^2 \cdot 5x^2 - x^2 \cdot 3x + x^2 \cdot \sqrt{3} \] Which simplifies to:\[ = x^5 + 5x^4 - 3x^3 + \sqrt{3}x^2 \]

Step 2: Integrate Each Term

Next, find the indefinite integral of each term in the expanded function:\[ \int x^5 \, dx = \frac{x^6}{6} \]\[ \int 5x^4 \, dx = 5 \cdot \frac{x^5}{5} = x^5 \]\[ \int -3x^3 \, dx = -3 \cdot \frac{x^4}{4} = -\frac{3x^4}{4} \]\[ \int \sqrt{3}x^2 \, dx = \sqrt{3} \cdot \frac{x^3}{3} = \frac{\sqrt{3}}{3}x^3 \]

Step 3: Combine the Integrated Terms

Now, sum all of the integrated terms together to form the general antiderivative:\[ F(x) = \frac{x^6}{6} + x^5 - \frac{3x^4}{4} + \frac{\sqrt{3}}{3}x^3 + C \] Here, \(C\) is the constant of integration.

Key Concepts

Antiderivative

In calculus, the antiderivative is essentially the reverse process of differentiation. When we say we are finding the antiderivative, we are searching for a function, let's call it \(F(x)\), whose derivative gives us the original function \(f(x)\). Think of it as asking: What was the function before it was differentiated?

The notation to express this process is typically an integral symbol \(\int\), and finding the antiderivative is also known as integrating a function. For example, if \(f(x) = x^2\), an antiderivative is \(F(x) = \frac{x^3}{3}\) because the derivative of \(\frac{x^3}{3}\) is \(x^2\).

It's important to note that there are infinitely many antiderivatives for any given function because of the constant of integration. This simply means any function \(F(x) + C\) (where \(C\) is any constant) will have the same derivative \(f(x)\).

Indefinite Integral

The indefinite integral is another way to describe the antiderivative of a function. When we refer to the "indefinite" part, we mean there is no specific interval that we're interested in. Instead, we're looking for a general expression for all possible antiderivatives of the function.

Mathematically, this is represented by \(\int f(x) \, dx = F(x) + C\), where \(f(x)\) is the function you're integrating, and \(F(x)\) represents the family of all antiderivatives with \(C\) being the constant of integration.

An indefinite integral provides the general formula for \(F(x)\), which includes all the possible vertical shifts represented by different values of \(C\). This formula can then be used to find specific values depending on initial or boundary conditions.

Polynomial Integration

Polynomial integration is a straightforward process in integral calculus, focusing on finding integrals of polynomial functions of the form \(ax^n\). The rule we use is called the Power Rule for Integration, which states that \(\int ax^n \, dx = \frac{ax^{n+1}}{n+1} + C\), provided \(n eq -1\).

For example, if you have \(\int x^5 \, dx\), applying the Power Rule gives you \(\frac{x^6}{6} + C\). Each term in a polynomial is integrated separately using this rule.

This method is particularly useful because polynomials frequently appear in calculus problems, and they break down easily into simpler parts to integrate. By applying the rule to each term, you build up the antiderivative for the entire polynomial.

Constant of Integration

The constant of integration, symbolized by \(C\), is a crucial element in calculus when working with indefinite integrals. When calculating an antiderivative, you are reversing the process of differentiation, and since differentiation "loses" any constant part of the function (because the derivative of a constant is zero), you need to add \(C\) to account for any constant that could have been part of the original function.

In practice, this means every time you find an indefinite integral, your result will include \(C\). For instance, if \(\int x^2 \, dx = \frac{x^3}{3} + C\), \(C\) represents any potential constant component of the original function before differentiation.

This constant is essential because it acknowledges the fact that there are infinitely many antiderivatives for a given derivative, differing only by a constant. When additional information is provided, such as a specific point the function passes through, \(C\) can be solved to find an exact antiderivative.