Problem 49 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \( C = 0 \)). \( \displaystyle \int x(x^2 - 1)^3 \, dx \)

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{(x^2 - 1)^4}{8} + C \) and is verified by graphing.

Step 1: Choose a substitution

First, identify a substitution that simplifies the integral. Let \( u = x^2 - 1 \). Then the differential \( du = 2x \, dx \) allows us to relate \( dx \) to \( du \). To isolate \( dx \), we solve for it: \( dx = \frac{du}{2x} \).

Step 2: Rewrite the integral in terms of \( u \)

Substitute \( u = x^2 - 1 \) and \( dx = \frac{du}{2x} \) into the integral. The integral becomes \( \int x(u)^3 \frac{du}{2x} \). Simplifying this gives \( \frac{1}{2} \int u^3 \, du \).

Step 3: Integrate with respect to \( u \)

Integrate \( \frac{1}{2} \int u^3 \, du \). The antiderivative of \( u^3 \) is \( \frac{u^4}{4} \). Therefore, the integral is \( \frac{1}{2} \times \frac{u^4}{4} + C = \frac{u^4}{8} + C \).

Step 4: Substitute back to \( x \) terms

Replace \( u \) back with \( x^2 - 1 \). This gives us \( \frac{(x^2 - 1)^4}{8} + C \). Since we want to verify, take \( C = 0 \). Thus, the antiderivative is \( \frac{(x^2 - 1)^4}{8} \).

Step 5: Check the solution by differentiating

Differentiate the antiderivative \( \frac{(x^2 - 1)^4}{8} \) with respect to \( x \). Using the chain rule, the derivative is \( \frac{1}{8} \times 4(x^2 - 1)^3 \times 2x = x(x^2 - 1)^3 \), which matches the original function.

Step 6: Graph both the function and its antiderivative

Use graphing software or a graphing calculator to plot both \( x(x^2 - 1)^3 \) and \( \frac{(x^2 - 1)^4}{8} \). The first graph should represent the rate of change of the second graph. Verify visually that the slopes on the antiderivative's graph correspond to the values of the original function at those points.

Key Concepts

Substitution Method

The substitution method is a technique used to simplify an integral by introducing a new variable to replace a function inside the integrand. This method is particularly useful when dealing with composite functions or when an integral includes a part that resembles the derivative of another part. Here's how it typically works:

• Identify a part of the integrand that can be replaced with a simpler expression. This often involves looking for a function that has its derivative also present in the integrand.
• Choose a substitution for this part, often setting it equal to a new variable, such as \( u \).
• Express the differential \( dx \) in terms of the new differential \( du \) to complete the substitution.

In the example given, the substitution \( u = x^2 - 1 \) was chosen because the derivative \( 2x \) is effectively present in the integral. This allowed the more complicated original integral to be rewritten in a simpler form \( \int u^3 \, du \), which is easier to integrate.

Antiderivative

The concept of an antiderivative is central to integral calculus. An antiderivative of a function is a function whose derivative is the original function. While differentiation answers the question of how a function changes, finding an antiderivative gives us the "parent function" before differentiation.

• To find the antiderivative of a function, we integrate it. The integration is often accompanied by a constant of integration, represented by \( C \), because differentiation of a constant results in zero.
• In the context of indefinite integrals, where there's no specific boundary, the result is an entire family of functions, all offset vertically by the constant \( C \).

In the given problem, after performing the substitution and simplification, the integral \( \int u^3 \, du \) was obtained. Its antiderivative \( \frac{u^4}{4} + C \) gives us back to the polynomial form, and when reshaped back to terms of \( x \), results in \( \frac{(x^2 - 1)^4}{8} + C \). For verification, \( C \) was set to zero.

Chain Rule

The chain rule is a fundamental principle in calculus that helps us differentiate composite functions. It describes how to take the derivative of a function that is made up of nested functions, where one function is inside another.

• Mathematically, if you have a function \( y = f(g(x)) \), then the derivative is given by the chain rule as \( f'(g(x)) \cdot g'(x) \).
• This method is crucial for correctly handling the derivatives when components of a function are expressed in terms of other variables.

In the solution given, the chain rule was applied during the verification step. After integrating and finding the antiderivative \( \frac{(x^2 - 1)^4}{8} \), we differentiated it to check correctness. Applying the chain rule allowed us to bring down the exponent and multiply by the inner function's derivative \( 2x \). This produced \( x(x^2 - 1)^3 \), confirming the correct antiderivative was found by matching it with the original function.