Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \int\left(\pi x^{3}+1\right)^{4} 3 \pi x^{2} d x $$

Step-by-Step Solution

Answer

\(\frac{(\pi x^3 + 1)^5}{5} + C\)

Step 1: Identify the Substitution

In this integral, we notice the presence of a composite function \((\pi x^3 + 1)^4\). This suggests using substitution. Let \(u = \pi x^3 + 1\).

Step 2: Differentiate Substitution Variable

Differentiate \(u\) with respect to \(x\). Thus, \(\frac{du}{dx} = 3\pi x^2\). We see that \(3\pi x^2 dx = du\).

Step 3: Substitute and Simplify

Replace \(\pi x^3 + 1\) with \(u\) and \(3\pi x^2 dx\) with \(du\) in the integral. The integral becomes \(\int u^4 du\).

Step 4: Integrate with Respect to u

The integral \(\int u^4 du\) is a standard power rule integral where \(\int u^n du = \frac{u^{n+1}}{n+1} + C\). Therefore, integrating, we get \(\frac{u^5}{5} + C\).

Step 5: Back Substitute in Terms of x

Replace \(u\) back with the original expression in terms of \(x\). We have \(u = \pi x^3 + 1\), so the integral becomes \(\frac{(\pi x^3 + 1)^5}{5} + C\).

Key Concepts

Substitution Method

The substitution method is a useful technique for simplifying the integration of complex functions. It involves replacing a complicated expression within an integral with a single variable, usually denoted as \( u \). This technique is particularly handy when dealing with composite functions, where one function is nested inside another.
For example, in the integral \( \int(\pi x^3+1)^4 3 \pi x^2 \, dx \), the function \( (\pi x^3 + 1)^4 \) makes it challenging to integrate directly. By letting \( u = \pi x^3 + 1 \), we simplify the integrand into a much more manageable form.
After substituting, it is important to express \( dx \) in terms of \( du \) so that the entire integral is expressed in terms of \( u \). In this exercise, differentiating \( u \) gives \( \frac{du}{dx} = 3\pi x^2 \), which implies that \( 3\pi x^2 \, dx = du \). By substituting \( u \) and \( du \) in place of the corresponding terms, the complex integral transforms into an easier one: \( \int u^4 \, du \).
This step simplifies the integration process significantly and is a cornerstone of the substitution method.

Power Rule in Integration

The power rule in integration is a simple and powerful tool to integrate expressions of the form \( \int x^n \, dx \). It states that if \( n eq -1 \), then:

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

where \( C \) is the constant of integration.
This rule is derived from the reverse process of differentiating power functions. The power rule can be applied after using substitution, as we simplify a complex integrand to a straightforward power of \( u \).
In the given problem, after applying substitution, we are left with the integral \( \int u^4 \, du \). Using the power rule of integration, we find that \( \int u^4 \, du = \frac{u^{5}}{5} + C \).
Understanding and practicing this rule is essential as it forms the basis of many more complicated integration problems.

Definite and Indefinite Integrals

Integrals are classified into two main types: definite and indefinite. An indefinite integral, such as \( \int x^n \, dx \), represents a family of functions and includes an arbitrary constant \( C \). This is because derivatives of a constant are zero, and thus one cannot determine the constant from the derivative alone. Indefinite integrals provide the most general form of the antiderivative.
The integral in the solved exercise, \( \int(\pi x^3+1)^4 3\pi x^2 \, dx \), is an indefinite integral. Therefore, its solution includes this constant of integration, denoted as \( C \), which represents all possible vertical shifts of the antiderivative.
In contrast, a definite integral computes the net area under the curve of a function between two specific points \( a \) and \( b \). It is written as \( \int_{a}^{b} f(x) \, dx \) and results in a fixed number, representing the accumulated quantity over an interval.
Understanding the distinction between these types of integrals is crucial in determining when to include the constant \( C \) and how to interpret the results of integration, whether as a general family of curves or as a specific numerical value.