Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check the result by differentiation. $$ \int\left(x^{2}-1\right)^{3}(2 x) d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\left(x^{2} - 1\right)^{3} \cdot 2x \, dx\) is \(\frac{1}{4}(x^{2}-1)^{4} + C\).

Step 1: Identify the composite function

The given expression in the integral, \(\left(x^{2} - 1\right)^{3}\cdot 2x\), is a composite function. Hence, we can use the substitution method to simplify it. We can identify \(u = x^{2} - 1\) and \(du = 2x dx\). The \(2x dx\) part of the initial integral resembles our \(du\), which hints us to make this substitution.

Step 2: Perform the substitution

Substituting \(u = x^{2} - 1\) and \(du = 2x dx\) into our integral, we have \(\int u^3 du \).

Step 3: Simplify the Integral

Do the actual integration. The integral of \(u^3\) is \(\frac{1}{4}u^4 + C\) using the power rule for integration. This gives us \(\frac{1}{4}u^4 + C\).

Step 4: Substitute back

To get back to terms of \(x\), substitute \(u\) back with \(x^2 - 1\) from Step 1. The result will be \(\frac{1}{4}(x^{2} - 1)^{4} + C\).

Step 5: Check the result with differentiation

Now, differentiate the function obtained. The derivative of \(\frac{1}{4}(x^{2} - 1)^{4} + C\) should give us \((x^{2}-1)^{3}(2 x)\). This checks out fine and our result was correct.

Key Concepts

Substitution Method in Integration

The substitution method, often used when dealing with composite functions in integration, simplifies the integral by choosing a new variable that represents a part of the original integrand. This approach involves identifying a function within the integrand and assigning it a new variable, usually denoted as \( u \). In essence, it transforms the integral into a simpler form that is easier to evaluate.
For the integral \( \int(x^{2}-1)^{3}(2x) dx \), we recognize a composite function and choose \( u = x^{2} - 1 \). The derivative of \( u \), which is \( du = 2x dx \), is also present in the integral, confirming our choice. After substituting \( u \) and \( du \) into the integral, we obtain \( \int u^3 du \), which is easier to integrate using the power rule. Substitution is particularly powerful for integrals where the derivative of the inside function is present in some form, allowing for a direct and often straightforward integration process.

Power Rule for Integration

The power rule for integration is a fundamental technique that allows us to solve integrals of polynomials. It states that the integral of \( x^n \), where \( n \) is any real number except -1, is given by \( \frac{x^{n+1}}{n+1} \), plus a constant of integration \( C \).
Applying this rule to our substituted integral \( \int u^3 du \), we integrate \( u \) raised to the third power. According to the power rule, the integral becomes \( \frac{1}{4}u^4 + C \). This simplification makes calculating indefinite integrals involving powers of \( x \) more manageable. The power rule is a crucial tool for quickly finding antiderivatives without resorting to more complex integration techniques.

Checking Integration by Differentiation

After finding an indefinite integral, it is good practice to verify the result by differentiation. This step is not only useful for checking our work but also reinforces the fundamental theorem of calculus, which posits that differentiation and integration are inverse processes.
In our example, after applying the power rule and substituting back, we obtained \( \frac{1}{4}(x^{2} - 1)^{4} + C \) as the antiderivative. To validate this result, we differentiate it with respect to \( x \). The derivative should yield the original integrand, \( (x^{2}-1)^{3}(2x) \). The differentiation process involves applying the chain rule, and if the original integrand is successfully retrieved, it confirms the accuracy of the integration process. This is a crucial final check in integration problems to ensure that our solution is indeed the correct antiderivative.