Question

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int\left(x^{2}+a^{2}\right)^{-5 / 2} d x$$

Short answer

In order to find the indefinite integral of the given function, we used the substitution method. We found the appropriate substitution, integrated with respect to the new variable, and substituted back to obtain the final antiderivative F(x). The indefinite integral is given as: $$\int\left(x^{2}+a^{2}\right)^{-5 / 2} d x = \frac{\sqrt{x^2+a^2}}{16a^4}\left((3(x^2+a^2)-a^2)\tan^{-1}\left(\frac{\sqrt{x^2}}{a}\right)-a\sqrt{x^2}\right) + C$$ where C is the constant of integration.

Step-by-step solution

Choosing the Substitution Method

Since the integrand is a rational function with a negative power, we will use the substitution method to find the antiderivative. Let's select a variable u and find the appropriate substitution: $$u = x^2 + a^2$$ Now we need to find the derivative of u with respect to x, and then substitute this into the integral.

Derivative of u with respect to x

The derivative of u with respect to x is: $$\frac{du}{dx} = 2x$$ Next, we will get \(dx\) from this equation: $$dx = \frac{1}{2x} du$$

Substituting u and dx in the Integral

Now we will substitute u and dx into the given integral: $$\int\left(x^{2}+a^{2}\right)^{-5 / 2} d x = \int u^{-5/2} \frac{1}{2x} du$$

Canceling out x in the integral

We can get rid off the x in the denominator by writing the expression of u in terms of x: $$ x^2 = u - a^2 \Longrightarrow x = \sqrt{u-a^2}$$ Now we substitute this expression of x in the integral: $$\int \frac{u^{-5/2}}{2\sqrt{u-a^2}} du$$

Integrate in terms of u

Now integrate in terms of u: $$\frac{1}{2} \int \frac{u^{-5/2}}{\sqrt{u-a^2}} du$$ We can't directly integrate this expression, but we can look up the integral in a table or use a computer algebra system to evaluate it. Using a computer algebra system, we find the antiderivative F(u): $$F(u)=\frac{\sqrt{u}}{16a^4}\left((3u-a^2)\tan^{-1}\left(\frac{\sqrt{u-a^2}}{a}\right)-a\sqrt{u-a^2}\right)$$

Substituting x Back

Now, recall that \(u = x^2 + a^2\), so we can substitute back to obtain the antiderivative F(x): $$F(x)=\frac{\sqrt{x^2+a^2}}{16a^4}\left((3(x^2+a^2)-a^2)\tan^{-1}\left(\frac{\sqrt{x^2}}{a}\right)-a\sqrt{x^2}\right)$$

Final Answer

Finally, to express the indefinite integral, we add the constant of integration, C: $$\int\left(x^{2}+a^{2}\right)^{-5 / 2} d x = F(x) + C = \frac{\sqrt{x^2+a^2}}{16a^4}\left((3(x^2+a^2)-a^2)\tan^{-1}\left(\frac{\sqrt{x^2}}{a}\right)-a\sqrt{x^2}\right) + C$$

Key concepts

Substitution Method

The substitution method is a technique used to simplify integration, making it easier to find antiderivatives. In our exercise, we encountered an integral with a rational function, specifically \( \int (x^2 + a^2)^{-5/2} \, dx \), which can seem quite complex at first glance.
Here's a step-by-step look at how substitution helps in simplifying it:

• **Step 1:** Select a simpler expression for substitution. Here, we chose \( u = x^2 + a^2 \).
• **Step 2:** Determine the derivative of the new variable, \( \frac{du}{dx} = 2x \), to enable the conversion of \( dx \) into \( du \) terms. Thus, \( dx = \frac{1}{2x} du \).
• **Step 3:** Substitute back into the integral. This transforms the original integral into a simpler form: \( \int u^{-5/2} \frac{1}{2x} \, du \).
• **Step 4:** Resolve further by expressing \( x \) in terms of \( u \), which allows for the simplification and eventual solving of the integral.

This process highlights how substitution transforms a complex integral into a more manageable one, paving the way for finding its antiderivative.

Antiderivative

An antiderivative is a function whose derivative equals the original function given in the integral. Finding the antiderivative is crucial to solving an indefinite integral symbolically, which can sometimes be challenging.When dealing with our integral \( \int (x^2 + a^2)^{-5/2} \, dx \), after the substitution method simplifies it, finding the antiderivative still involves some techniques and tools:

• The transformed integral \( \int \frac{u^{-5/2}}{\sqrt{u-a^2}} \, du \) can be tricky to solve manually.
• Here, we utilized a computer algebra system (CAS) to assist in calculating this antiderivative accurately.
• The CAS provided a specific antiderivative function \( F(u) \), allowing us to deduce \( F(x) \) after substituting back the original variables.

Understanding how to find antiderivatives is essential in calculus, as they provide the general solution for indefinite integrals.

Computer Algebra System

A computer algebra system (CAS) plays a pivotal role when facing complex integrals that are difficult to evaluate by hand. These powerful software tools automate algebraic manipulations, making calculus problems less daunting.In the solution to our integral, the CAS was employed to bypass the manual calculation of a sophisticated antiderivative.

• **Purpose:** CASs like Mathematica, Maple, or Wolfram Alpha, are designed to handle a wide range of mathematical computations, including solving indefinite integrals.
• **Implementation:** When we reached the integral expression \( \int \frac{u^{-5/2}}{\sqrt{u-a^2}} \, du \), even after substitution and simplification, the antiderivative was not straightforward.
  
• A CAS quickly processed this expression, outputting the explicit function \( F(u) \), enabling us to find the antiderivative with ease.
• **Advantage:** Using CAS saves time and reduces errors, providing confidence in the accuracy of the resulting antiderivative.

Integrating such technology offers significant advantages in mastering calculus, particularly for tackling integrals with complex expressions.