Question

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. $$ \int \tan ^{3}(1-x) d x $$

Short answer

First, rewrite the integral as \(\int \tan (1-x) *(\sec ^2(1-x) - 1) dx\). After substitution, you get \(-\tan (u) + u + C\), and by substituting \(x\) back, the final result is \(-\tan (1-x) + (1-x) + C\). To draw the antiderivatives, consider the values of \(C\) as \(C_1\) and \(C_2\) and plot the functions against \(x\).

Step-by-step solution

Compute the Integral

To find the integral of \(\tan ^{3}(1-x) dx\), apply integral calculus techniques. Rewriting the integral as \(\int \tan (1-x) *\tan ^2(1-x) dx\) can help in calculations. Using pythagorean identity, the latter part of the function can be rewritten as \(\sec ^2(1-x) - 1\). Now, the function can be integrated more easily by substitution.

Perform the substitution

Let \( u = 1-x\) and \( du = -dx\). On substituting, the integral transforms into \(-\int \tan (u) *(\sec ^2(u) - 1) du\). Upon performing this integration, the result is \(-\tan (u) + u + C\). Convert back into the variable \(x\) to get \(-\tan (1-x) + (1-x) + C\) as the final result.

Graph antiderivatives

Since the constant of integration, \(C\), can be any real number, two suitable values must be selected for further plotting. Let's consider \(C_1\) and \(C_2\) as two arbitrary constants. Thus, the two antiderivatives are \(-\tan (1-x) + (1-x) + C_1\) and \(-\tan (1-x) + (1-x) + C_2\). Creating a graph with \(x\) as the independent variable and the function \(-\tan (1-x) + (1-x) + C\) as the dependent variable, with \(C_1\) and \(C_2\) as the constants, will draw two distinct curves that represent the antiderivatives.

Key concepts

Antiderivatives

Antiderivatives, or indefinite integrals, are the reverse operation of differentiation. Just as subtraction undoes addition, finding an antiderivative undoes the process of taking a derivative. In essence, if you have a function f(x), an antiderivative F(x) is a function whose derivative is f(x), so that F'(x) = f(x).

When calculating antiderivatives, one must include a constant of integration, C, because differentiation eradicates constants. This is why antiderivatives are not unique: different values of C lead to different antiderivative functions, which all represent the same family of curves on a graph, each one shifted vertically by the constant C.

In the given exercise, after finding the antiderivative of \(\tan^{3}(1-x)\), we acquire \( -\tan (1-x) + (1-x) + C \), which is a general expression for an entire set of curves, differentiated from one another by varying the constant C.

Trigonometric Integration

Trigonometric integration involves integrating functions with trigonometric identities, such as sine, cosine, tangent, and their respective inverses. It often requires the utilization of various identities to simplify the integral into a more manageable form.

For example, Pythagorean identities are frequently used to convert between trigonometric functions. This is exactly what was done in the step-by-step solution when the term \(\tan^2(1-x)\) was transformed to \(\sec^2(1-x) - 1\), streamlining the integration process. Such tricks are essential for tackling complex trigonometric integrals and are often needed prior to other integration methods, like substitution or integration by parts.

Substitution Method

The substitution method in integral calculus is akin to finding the reverse of the chain rule in differentiation. It is especially useful for functions that are compositions of simpler functions whose integration is not straightforward.

The method involves choosing a new variable, usually denoted as u, which simplifies the original integral. Often, a function and its derivative are present in the integrand, prompting a natural choice for substitution. After substituting, the integral becomes easier to manage and more recognizable.

In the exercise provided, setting \( u = 1-x \) simplifies the integral significantly, transforming \(\int \tan ^{3}(1-x) dx\) into a simpler form that can be readily integrated. Remember, the differential \(du\) must also replace the original 'dx', hence why \(du = -dx\), effectively revising the whole integral in terms of u.

Computer Algebra System

A computer algebra system (CAS) is a software tool that facilitates a wide range of mathematical computations, including calculus operations like differentiation and integration. CAS tools can handle symbolic operations, meaning they can perform algebraic operations without needing numerical values, which is particularly helpful for integrals and derivatives.

In this exercise, a CAS would be used to evaluate the integral of \(\tan^{3}(1-x)\). The power of a CAS lies in its ability to quickly and accurately perform complex mathematical operations that would be incredibly time-consuming to do by hand. Furthermore, a CAS can plot the resulting antiderivatives for different values of the constant of integration, providing a clear visual understanding of how the constant affects the function's graph.