Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate the indicated indefinite integrals. $$ \int\left(x^{2}+x\right) d x $$

Step-by-Step Solution

Answer

\( \frac{x^3}{3} + \frac{x^2}{2} + C \)

Step 1: Identify the Integral Expression

We are given a function to integrate: \( \int (x^2 + x) \ dx \). This is the expression we need to find the indefinite integral for.

Step 2: Use the Sum Rule of Integration

The sum rule allows us to integrate the sum of two functions separately. Thus, we can rewrite the integral as:\[\int (x^2 + x) \, dx = \int x^2 \, dx + \int x \, dx\]

Step 3: Apply the Power Rule to \(x^2\)

To integrate \(x^2\), use the power rule which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Applying it here:\[\int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}\]

Step 4: Apply the Power Rule to \(x\)

Similarly, apply the power rule to \(x\), recognizing \(x\) as \(x^1\):\[\int x \, dx = \int x^1 \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2}\]

Step 5: Combine Integrals and Add Constant of Integration

Combine the results from Steps 3 and 4 to obtain the final integral result:\[\int (x^2 + x) \, dx = \frac{x^3}{3} + \frac{x^2}{2} + C\]where \(C\) is the constant of integration.

Key Concepts

Sum Rule of Integration

The sum rule of integration is both simple and incredibly powerful. This rule states that the integral of a sum of functions is equal to the sum of their individual integrals. For instance, if we have two functions, say \( f(x) \) and \( g(x) \), the sum rule tells us:

• \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \)

This rule is crucial because it allows us to break down more complex integrals into easier, manageable parts.
It helps us focus on one function at a time, making the process of integration less daunting.
In the example given, \( \int (x^2 + x) \, dx \) was separated into two distinct integrals: \( \int x^2 \, dx \) and \( \int x \, dx \).
By addressing each piece separately, we simplify the problem significantly.
Thus, the sum rule streamlines the process of finding indefinite integrals, especially when dealing with polynomial expressions.

Power Rule of Integration

The power rule of integration is one of the core tools in calculus for finding indefinite integrals. This rule is particularly useful when integrating polynomials, as it provides a straightforward method to find antiderivatives. The power rule states:

• If \( n eq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \( C \) is the constant of integration, and \( n \) is the exponent of the variable.
This rule requires us to increase the exponent by one and then divide by this new exponent.
In the solution to our example:

• For \( x^2 \), using the power rule, \( \int x^2 \, dx = \frac{x^{3}}{3} \).
• For \( x^1 \), it becomes \( \int x \, dx = \int x^1 \, dx = \frac{x^2}{2} \).

This simple yet effective tool helps us manage integration tasks by providing a quick method to find general antiderivatives of polynomial functions.

Constant of Integration

When we solve indefinite integrals, we encounter an interesting component known as the constant of integration, denoted by \( C \). This constant arises because indefinite integrals encompass a family of functions.

• Why a constant? Because differentiation of any constant is zero, when finding indefinite integrals, we cannot determine an exact value for this constant from the integral alone.

Therefore, \( C \) represents all possible constants that can be added to an antiderivative to make it a particular solution.
In our example, after applying the sum and power rules, we combined the calculated integrals of each term:

• \( \frac{x^3}{3} + \frac{x^2}{2} + C \)

This final expression includes \( C \) to reflect that any of a family of functions can serve as the antiderivative of \( x^2 + x \).
Understanding the constant of integration is vital as it emphasizes the need for additional context or conditions to find specific solutions among the infinite possibilities.