Question

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int\left(a^{2}-x^{2}\right)^{-2} d x$$

Short answer

Question: Evaluate the indefinite integral of the function $$\left(a^{2}-x^{2}\right)^{-2}$$ with respect to x, where "a" is a positive real number. Answer: $$\int\left(a^{2}-x^{2}\right)^{-2} dx = \frac{1}{2}\cdot(a^2 - x^2)^{-1} + C$$

Step-by-step solution

Substitution

Let's make a substitution to simplify the given function. We have$$\left(a^{2}-x^{2}\right)^{-2}.$$Let's say$$u = a^2 - x^2.$$Now, differentiate both sides with respect to x:$$\frac{du}{dx} = -2x.$$To get$$dx = \frac{du}{-2x}.$$Now, we can substitute u and dx back in the original integral:$$\int\left(a^{2}-x^{2}\right)^{-2} dx = \int u^{-2} \frac{du}{-2x}.$$

Canceling terms and Integrating

Now, we'll cancel the term "x" from the integral:$$\int u^{-2} \frac{du}{-2x} = \frac{-1}{2} \int u^{-2} du.$$Now let's integrate u with respect to u:$$\frac{-1}{2} \int u^{-2} du = \frac{-1}{2} \left[\frac{u^{-1}}{-1}\right] + C = \frac{1}{2} \cdot u^{-1} + C.$$

Substituting back x

Now, substitute back the value of u in terms of x:$$\frac{1}{2} \cdot u^{-1} + C = \frac{1}{2}\cdot(a^2 - x^2)^{-1} + C.$$The final result is:$$\int\left(a^{2}-x^{2}\right)^{-2} dx = \frac{1}{2}\cdot(a^2 - x^2)^{-1} + C.$$

Key concepts

Substitution Method

The substitution method is a powerful tool for solving integration problems. In our exercise, we simplified the integral by using substitution, which can make complex problems much simpler.
When you perform a substitution, you replace a complicated expression with a new variable. In this example, we transformed the expression \((a^2 - x^2)^{-2}\) into something more manageable by letting \( u = a^2 - x^2 \).
This process helps to reduce the complexity of the original integral.

• Identify the inner function you want to replace with a variable, typically something repetitive or difficult to integrate directly.
• Define your new variable \( u \), based on this function. In our example, it was \( u = a^2 - x^2 \).
• Find \( \frac{du}{dx} \) and solve for \( dx \) to substitute in the integral.
• Perform the substitution in the integral to convert it into a simpler form.

By replacing complex parts of the function with \( u \), we simplify integration while maintaining the relationship. It's like paving the way through a complicated maze.

Integration Techniques

Integration techniques are essential in calculus for evaluating an integral. A technique is selected based on the form of the function.
In our solution, once we used substitution successfully, the remaining expression \( \int u^{-2} \) is straightforward to integrate. Here are some crucial techniques to keep in mind:

• **Basic Power Rule:** If you need to integrate \( u^n \), you use \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) provided \( n eq -1 \).
• **Substitution:** As featured in our exercise, this is especially useful when the derivative of a part of the integrand is present elsewhere in the integrand.
• **Partial Fractions:** This technique is useful when dealing with rational expressions, especially when the degree of the numerator is less than the degree of the denominator.
• **Integration by Parts:** Ideal when the integrand is a product of two functions, based on the rule \( \int u v' \, dx = u v - \int v u' \, dx \).

Selecting the right technique often requires practice and understanding of each method's scope and limitations. For our case, using substitution streamlined the integration, converting the function into a more manageable form.

Algebraic Simplification

Algebraic simplification in integration involves rearranging and reducing expressions to their simplest form.
After performing a substitution and separating the terms, we frequently encounter integrals that seem solvable yet are cluttered. Simplification helps:

• Cancel out terms. For instance, reducing \( \frac{du}{-2x} \) in the integral by cancelling where necessary.
• Rewriting expressions to match known integral forms simplifies the problem.
• Introducing constants, such as dividing by 2 or factoring out simple arithmetic multipliers, streamlines calculations and can save steps.

In our problem, by simplifying \(\int u^{-2} \frac{du}{-2x}\) to \(\frac{-1}{2} \int u^{-2} du\), we reduced the steps needed to integrate. Simplification clears the clutter, revealing straightforward paths to solve integrals. It turns complex expressions into simple, recognizable forms.