Problem 12 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int x\left(x^{2}+3\right)^{2} d x $$

Step-by-Step Solution

Answer

\(\frac{1}{6}(x^2 + 3)^3 + C\)

Step 1: Choose a Substitution

To simplify the integral, we'll use a substitution method. We choose substitution \( u = x^2 + 3 \). Then, the derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 2x \). Rearranging gives us \( du = 2x \ dx \), or \( \frac{1}{2}du = x \ dx \). This substitution will simplify the integral.

Step 2: Rewrite the Integral

Substitute \( u = x^2 + 3 \) and \( x \, dx = \frac{1}{2} \, du \) into the integral. The integral becomes: \[ \int x(x^2 + 3)^2 \, dx = \int \frac{1}{2} u^2 \, du \]Factor out the constant: \[ \frac{1}{2} \int u^2 \, du \]

Step 3: Integrate with Respect to u

To find \( \int u^2 \, du \), use the power rule for integration which states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Applying this here gives:\[ \frac{1}{2} \cdot \frac{u^3}{3} + C = \frac{u^3}{6} + C \]

Step 4: Substitute Back to x

Since \( u = x^2 + 3 \), substitute back into the expression for the integral:\[ \frac{1}{6} (x^2 + 3)^3 + C \]

Step 5: Differentiate to Check

Differentiate the result \( \frac{1}{6} (x^2 + 3)^3 + C \) with respect to \( x \) to verify our solution. Apply the chain rule:\[ \frac{d}{dx} \left( \frac{1}{6} (x^2 + 3)^3 \right) = \frac{1}{6} \cdot 3(x^2 + 3)^2 \cdot 2x = x(x^2 + 3)^2 \]This shows that our antiderivative is correct because it matches the original integrand.

Key Concepts

Substitution Method

The substitution method, sometimes called the "u-substitution," is a powerful technique in integration. It helps to transform an integral into a simpler form, making it easier to solve. In our original problem, we have the integral \( \int x(x^2 + 3)^2 \, dx \). The expression \((x^2 + 3)\) is complex, so we simplify it by letting \( u = x^2 + 3 \). This change of variable simplifies the complex expression, which can then be integrated more easily.
Next, we find the derivative of our substitution, which is \( \frac{du}{dx} = 2x \), and rewrite it as \( du = 2x \, dx \) or \( \frac{1}{2} du = x \, dx \). By substituting these expressions into the integral, the problem is transformed into an expression in terms of \( u \), namely \( \int \frac{1}{2} u^2 \, du \). This is a much simpler integral to solve, demonstrating why substitution is such a useful technique.

Power Rule for Integration

Integrating functions uses various rules, one of the most fundamental being the power rule for integration. This rule simply extends the idea of finding antiderivatives for expressions of the form \( u^n \). Specifically, if you need to integrate \( \int u^n \, du \), the power rule tells us this is equal to \( \frac{u^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
In our problem, since we transformed the integral into \( \int \frac{1}{2} u^2 \, du \), applying the power rule allows us to find \( \int u^2 \, du = \frac{u^3}{3} \). Don't forget to factor out and adjust constants as needed: here, we factored out \( \frac{1}{2} \), resulting in \( \frac{1}{2} \cdot \frac{u^3}{3} = \frac{u^3}{6} \). This method of solving is not only simple but also systematic.

Indefinite Integrals

An indefinite integral, sometimes called an antiderivative, represents a family of functions. These functions differ by a constant, and they "undo" the process of differentiation. In contrast to a definite integral, which calculates an area under a curve between two bounds, an indefinite integral does not have specific limits of integration.
In the original exercise, finding an indefinite integral involved \( \int x(x^2 + 3)^2 \, dx \), where we wanted an expression for all potential antiderivatives. After substitutions and integration, the solution is given by \( \frac{(x^2 + 3)^3}{6} + C \). The "\( + C \)" at the end of the expression signifies that this is one of many possible solutions. The constant \( C \) accounts for all vertical shifts of the antiderivative graph.

Chain Rule

The chain rule is primarily known as a differentiation technique, but it's crucial for verifying our integration results. It helps find the derivative of a composite function, ensuring your antiderivative is correct. Applying the chain rule means differentiating an outer function while multiplying that by the derivative of the inner function.
In our exercise's context, after achieving the integrated form \( \frac{1}{6}(x^2 + 3)^3 + C \), the chain rule is applied during differentiation to check our work. Here, we treat \((x^2+3)^3\) as a composite function, with the outer function being \( (x^2 + 3)^3 \) and the inner function \( (x^2 + 3) \). Differentiation using the chain rule confirms we return to the original form \( x(x^2 + 3)^2 \). This validation step is crucial to confirm the integration process was performed correctly.