Problem 11 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int x\left(x^{2}-4\right)^{7 / 2} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{9} (x^2 - 4)^{9/2} + C \).

Step 1: Choose a Substitution

Let's use the substitution method. Set \( u = x^2 - 4 \). Then compute the derivative: \( du = 2x \, dx \). This suggests \( x \, dx = \frac{1}{2} \, du \).

Step 2: Substitution in the Integral

Substitute \( u = x^2 - 4 \) into the integral and \( x \, dx = \frac{1}{2} \, du \). The integral becomes: \( \int \left(u^{7/2}\right) \cdot \frac{1}{2} \, du = \frac{1}{2} \int u^{7/2} \, du \).

Step 3: Integrate with Respect to u

Integrate \( \frac{1}{2} \int u^{7/2} \, du \). The antiderivative is \( \frac{1}{2} \cdot \frac{2}{9} u^{9/2} + C = \frac{1}{9} u^{9/2} + C \).

Step 4: Substitute Back

Replace \( u \) with \( x^2 - 4 \). The integral becomes \( \frac{1}{9} (x^2 - 4)^{9/2} + C \).

Step 5: Differentiate the Result to Check

Differentiate \( \frac{1}{9} (x^2 - 4)^{9/2} + C \) with respect to \( x \). Using the chain rule: \( \frac{d}{dx} \left( \frac{1}{9} (x^2 - 4)^{9/2} \right) = \frac{1}{9} \cdot \frac{9}{2}(x^2 - 4)^{7/2} \cdot 2x = x(x^2 - 4)^{7/2} \), which matches the original integrand.

Key Concepts

Substitution Method

The substitution method is a technique used in integration to simplify complex expressions, making them easier to solve. The idea is to select a new variable, usually denoted by \( u \), to replace a complicated part of the integrand. For the given exercise, we choose \( u = x^2 - 4 \).
This substitution is strategic because it simplifies the expression \( (x^2 - 4)^{7/2} \) when differentiated. By finding the derivative of \( u \), which is \( du = 2x \, dx \), we relate the new variable back to the original variable. This transformation allows the integrand to be rewritten in terms of \( u \) and \( du \), often resulting in a more straightforward integration.

Helpful tips:

• Choose \( u \) so its derivative appears in the integrand.
• Note the relationships between \( dx \), \( du \), and other expressions.

With our choice, we get \( x \, dx = \frac{1}{2} \, du \), leaving the integral as \( \frac{1}{2} \int u^{7/2} \, du \). This substitution significantly simplifies further integration steps.

Antiderivative

The antiderivative is the reverse operation of differentiation. It involves finding a function whose derivative is the given integrand. When faced with integration, the goal is often to find this antiderivative.
In our exercise, after substitution, the task simplifies to finding the antiderivative of \( u^{7/2} \). Thankfully, there is a straightforward rule: for \( u^n \), the antiderivative is \( \frac{u^{n+1}}{n+1} \) provided \( n eq -1 \).

Applying this to \( u^{7/2} \), we get:

• The antiderivative of \( u^{7/2} \) is \( \frac{2}{9} u^{9/2} \).
• Don't forget to multiply by \( \frac{1}{2} \) due to our earlier substitution, resulting in \( \frac{1}{9} u^{9/2} \).

This result represents the antiderivative in terms of \( u \), and substituting back \( x^2 - 4 \) completes the process.

Chain Rule

The chain rule is a fundamental rule of differentiation used to handle composite functions. Whenever you have a function inside another, like \( (x^2 - 4)^{9/2} \), the chain rule helps us differentiate it correctly.
In mathematical terms, if \( f(g(x)) \) is our function, then the derivative is \((f'(g(x)) \cdot g'(x) \). In the exercise, this rule is crucial for verifying the solution. To differentiate \( \frac{1}{9} (x^2 - 4)^{9/2} \), first apply the rule to \( (x^2 - 4)^{9/2} \).

• First, differentiate the outer function: \( \frac{9}{2}(x^2 - 4)^{7/2} \).
• Then, multiply by the derivative of the inner function: \( 2x \).

Combining these results and simplifying confirms that the chain rule produces the original integrand \( x(x^2 - 4)^{7/2} \), validating the antiderivative and substitution used earlier.

Differentiation

Differentiation is the mathematical process of finding how a function changes as its input changes. It is the cornerstone of calculus and essential for confirming the correctness of antiderivatives. In our problem, once we find an antiderivative, it's crucial to differentiate it to ensure accuracy.
The goal is to recover the original function, \( x(x^2 - 4)^{7/2} \), after substituting and integrating. Applying differentiation:

• Use the chain rule for every complicated term, checking each step.
• Ensure that derivatives lead back to the stated integrand.
• Test your final expression against the original problem.

When differentiating \( \frac{1}{9} (x^2 - 4)^{9/2} \), careful application of the chain rule returns \( x(x^2 - 4)^{7/2} \). This confirms the integrity of your integration, substitution, and algebraic steps. Differentiation acts as a powerful tool to verify integration solutions.