Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \int(\sqrt{2} x+1)^{3} \sqrt{2} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{(\sqrt{2}x + 1)^4}{4} + C \).

Step 1: Understanding the Integral

We need to find the indefinite integral of \( \int(\sqrt{2} x+1)^{3} \sqrt{2} \, dx \). To make this process easier, we can apply substitution.

Step 2: Choose a Suitable Substitution

Let's use substitution to simplify this integral. Set \( u = \sqrt{2}x + 1 \). Then, the derivative \( du = \sqrt{2} \, dx \). This implies that \( dx = \frac{du}{\sqrt{2}} \).

Step 3: Substitute and Rewrite the Integral

Substitute \( u = \sqrt{2} x + 1 \) and \( dx = \frac{du}{\sqrt{2}} \) into the integral:\[ \int u^3 \cdot \sqrt{2} \cdot \frac{du}{\sqrt{2}} = \int u^3 \, du \]

Step 4: Integrate the Simplified Expression

Now, integrate \( u^3 \, du \) to get \( \frac{u^4}{4} + C \), where \( C \) is the constant of integration:

Step 5: Substitute Back the Original Variable

Replace \( u \) with \( \sqrt{2} x + 1 \) to get the original variable back in the expression:\[ \frac{(\sqrt{2}x + 1)^4}{4} + C \]

Step 6: Final Result

The integral \( \int(\sqrt{2} x+1)^{3} \sqrt{2} \, dx \) evaluates to \( \frac{(\sqrt{2}x + 1)^4}{4} + C \).

Key Concepts

Substitution Method

The substitution method is an integration technique that simplifies complex integrals by substituting a part of the integral with a single variable. In the given exercise, let's understand its application. Using substitution helps to transform a challenging integral into a more straightforward one.
In this problem, we start by recognizing a function within the integral that can be set as a new variable. We choose to set \( u = \sqrt{2} x + 1 \). This simplifies the integration process significantly.
After setting \( u \), the derivative becomes \( du = \sqrt{2} \, dx \), which implies that \( dx = \frac{du}{\sqrt{2}} \). This step is crucial to rewrite the integral in terms of \( u \) and \( du \) rather than \( x \) and \( dx \). The original integral \( \int ( \sqrt{2}x + 1 )^3 \sqrt{2} \, dx \) then transforms into \( \int u^3 \, du \). This simplified form is much easier to integrate, showing how powerful substitution can be for solving integrals.

Integration Techniques

Integration techniques are essential tools used to solve integrals, especially when they appear in complex forms.
There are several methods, but in this exercise, substitution is the chosen technique due to its effectiveness in simplifying the integral. By changing the variable, the integral form, originally difficult, was turned into a basic polynomial integral.
Once the problem is in the form of \( \int u^3 \, du \), the integration technique becomes straightforward. The polynomial rule states that \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where \( n \) is a constant exponent. Applying this to our case with \( n=3 \), we get \( \int u^3 \, du = \frac{u^4}{4} + C \). Thus, the integral is simplified using the basic polynomial integration rule, illustrating how substitution effectively simplifies the process.

Constant of Integration

Every indefinite integral comes with a constant of integration, represented by \( C \).
Let's explore its importance in the context of this exercise. When integrating, the process determines a family of functions rather than a specific one. Therefore, adding \( C \) accounts for any constant that could be part of the antiderivative.
The integral \( \int u^3 \, du = \frac{u^4}{4} + C \) includes this constant, revealing that there are infinitely many solutions, each differing by a constant. When we substitute back our original variable, the expression becomes \( \frac{(\sqrt{2}x + 1)^4}{4} + C \). No matter the specific scenario, \( C \) ensures the integrity of the solution by acknowledging that any constant may have been present in the antiderivative's original form.