Problem 10 - Essential Calculus

Question

Find the indefinite integral and check the result by differentiation. $$ \int x\left(4 x^{2}+3\right)^{2} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(x(4x^{2}+3)^{2}\) is \((1/24)(4x^{2}+3)^{3} + C\).

Step 1: Choose an appropriate substitution

Choose \(u\) as \(4x^{2}+3\). This choice is made because the derivative of \(u\), which is \(8x\), is present within the integral. Remember that in a substitution, you want to choose a \(u\) whose derivative is entirely or partly inside the integral.

Step 2: Find du and substitute in the integral

Find \(du\) as \(du = 8xdx\). Divide by 8 to isolate \(dx\) in terms of \(du\), that is \(dx = du / 8x\). Substitute \(u\) and \(dx\) into the original integral, leading to a new integral \(\int u^{2} (du / 8)\).

Step 3: Simplify the integral and solve

Pull out the constant 1/8 out of the integral to simplify it, obtaining \(1/8\int u^{2} du\). Now, solve this integral obtaining \(1/8 * (1/3)u^{3} + C\), where C is the constant of integration.

Step 4: Substitute back and simplify the result

Substitute back \(u = 4x^{2}+3\) into the solution. The solution to the integral is then \((1/24)(4x^{2}+3)^{3} + C\).

Step 5: Check the result by differentiation

Differentiating our result \((1/24)(4x^{2}+3)^{3} + C\) should return the function to be integrated, i.e., \(x(4x^{2}+3)^{2}\). The derivative verifies the integral.

Key Concepts

Substitution Method

The substitution method is a key technique in calculus used to simplify integration. This method revolves around the idea of substituting a part of the integral to streamline the computation. By choosing an appropriate substitution, you can transform a complex integral into a simpler one, which is easier to evaluate.

In this exercise, the function within the integral, \( \int x(4x^{2}+3)^{2} dx \), is ripe for substitution. The expression \( 4x^{2} + 3 \) is a prime candidate because its derivative, \( 8x \), is already part of the integral.

Here's how you do it:

• Define a new variable \( u = 4x^{2} + 3 \).
• Find the differential \( du = 8xdx \).
• Solve for \( dx \) in terms of \( du \, \text{and}\, x \): \( dx = du / 8x \).

These steps transform the original integral into a much simpler form, \( \int u^{2} (du / 8) \), allowing you to proceed with the integration in a straightforward manner.

Substitution is essential because it reduces the complexity of integrals, making them more manageable for students and mathematicians alike.

Integration Techniques

Integration techniques like the substitution method play a crucial role in evaluating indefinite integrals. The goal is to find the antiderivative, a function whose derivative gives back the original function being integrated.

The typical steps for handling a challenging integral involve:

• Identifying parts of the integral that can be replaced or transformed.
• Substituting variables to simplify the problem, often using algebraic rearrangements.
• Simplifying the integral to the point where basic integration rules can be applied.

In this exercise, after substituting, the integral \( \int u^{2} du \) becomes straightforward to solve. The integration result is \( (1/3)u^{3} \), simplified as \( (1/24)(4x^{2}+3)^{3} + C \) after substituting back the original variables.

This simplification illustrates the power of integration techniques, allowing for the resolution of seemingly complex problems with simpler steps. Proper application of these techniques is key to mastering calculus.

Differentiation

Differentiation is like the mirror image of integration in calculus, often used to verify integration results. After finding an indefinite integral, it's crucial to differentiate the result to ensure it matches the original function that was integrated.

In this specific problem, we found the integral to be \((1/24)(4x^{2}+3)^{3} + C\). The next step is differentiation:

• Differentiate with respect to \(x\).
• Use the chain rule correctly as the function involves a composite term \((4x^{2}+3)^{3}\).

Differentiating \((1/24)((4x^{2}+3)^{3})\) gives the original function \(x(4x^{2}+3)^{2}\), thus validating the integration process.

This practice of checking your work through differentiation is important, as it confirms the accuracy of your integration and builds confidence in solving calculus problems. Differentiation thus ensures the reliability and correctness of the indefinite integrals computed.