Question

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. $$ \int \sec ^{5} \pi x \tan \pi x d x $$

Short answer

The antiderivative of the given expression is \( F(x) = \frac{(\sec \pi x)^6}{6\pi} + C \). Graphs of the antiderivatives for \( C = 0 \) and \( C = 1 \) would show the different possible antiderivatives of the given function.

Step-by-step solution

U-Substitution

We'll start with a u-substitution. Let \( u = \sec \pi x \), this makes the derivative \( du = \sec \pi x \tan \pi x \pi dx \). So the integral becomes \( \frac{1}{\pi} \int u^5 du \).

Integration

Now we can integrate \( \frac{1}{\pi} \int u^5 du \). Using power rule for integration, which states that \( \int x^n dx = \frac{1}{n+1}x^{n+1} + C \), where C is an integration constant, we get \( \frac{u^6}{6\pi} + C \).

Substituting Back

Replace u with \( \sec \pi x \). So the antiderivative will be \( F(x) = \frac{(\sec \pi x)^6}{6\pi} + C \).

Graph

Finally, we graph the antiderivatives for two different values of the constant of integration. For instance, we can pick \( C = 0 \) and \( C = 1 \). Graphs of \( F(x) = \frac{(\sec \pi x)^6}{6\pi} \) and \( F(x) = \frac{(\sec \pi x)^6}{6\pi} + 1 \) will give the required visual representation.

Key concepts

U-Substitution

Understanding U-Substitution is critical when dealing with complex integrals, especially those involving products or composite functions. Essentially, it's a method for simplifying an integral by substituting part of the integral with a new variable, typically denoted by 'u'. This method often transforms a difficult integral into a more manageable form.

As seen in the exercise solution, U-Substitution involves choosing a 'u' that is a function within the integral that, when differentiated, resembles another part of the integrand. In our example, we selected u to be \( \text{sec} \theta \), specifically u = \text{sec} \theta \text{x}. The next step is to find the differential of 'u', denoted as 'du'. In this case, the derivative of \text{sec} \theta \text{x} with respect to 'x' is \text{sec} \theta \text{x} \tan \theta \text{x}, which we have in the original integral. This resemblance signifies a correct U-Substitution.

This strategy simplifies the integral from its original trigonometric form into a polynomial form, which is generally easier to integrate. It's important to remember to convert all the 'x' terms to 'u' terms during the substitution, including the differential 'dx', and that the bounds of integration (if it's a definite integral) will also change as a consequent of the substitution.

Antiderivative

The concept of an antiderivative is crucial in calculus. It is another term for the indefinite integral and represents the inverse operation to differentiation. When you take the antiderivative of a function, you're essentially finding the original function whose derivative gives you the integrand. The antiderivative comes with a constant of integration, often represented as 'C', which encompasses all possible constants that could be added to the function without changing its derivative.

In the given problem, the process of finding the antiderivative involves integrating a power of a trigonometric function. After applying U-Substitution, we obtain a new integral in terms of 'u', which is a simple polynomial form. The antiderivative of this polynomial is straightforward to obtain using the power rule for integration.

The last step is to translate the solution back in terms of the original variable 'x' to complete the antiderivation process. It's important to note that we need to consider the constant of integration 'C', which allows us to graph multiple versions of the antiderivative by selecting different values for 'C'. This constant represents all possible vertical shifts of the graph, each giving a valid antiderivative of the original function.

Power Rule for Integration

One of the fundamental rules in calculus for finding antiderivatives is the Power Rule for Integration. This rule states that for any real number 'n', different from -1, the integral of x^n with respect to 'x' is \( \frac{1}{n+1}x^{n+1} \) plus a constant of integration 'C'. Simply put, you add one to the exponent and divide by the new exponent.

Applied to our example, after the U-Substitution, we worked with u^5. Following the power rule, we added one to the power to get u^6 and divided by the new exponent, yielding \( \frac{1}{6}u^{6} \). Don't forget to adjust for any constants that were part of the integral before substitution, as we did with the \( \frac{1}{\pi} \) that was factored out initially.

This rule is immensely powerful because it allows us to integrate any polynomial easily. However, keep in mind it does not apply to functions with an exponent of -1 because this leads to the natural logarithm function, which has its own integration rules. Practicing the Power Rule with a variety of polynomials can build a strong base for understanding more complex integrations involving other functions.