Question

(A) find the indefinite integral in two different ways. (B) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (C) Verify analytically that the results differ only by a constant. $$ \int \sec ^{4} 3 x \tan ^{3} 3 x d x $$

Short answer

The antiderivative of the function \( \sec^4 3x \tan^3 3x \) can be calculated in two ways: (1) \( \frac{1}{4}(\sec^4 3x) + C \) using substitution; and a second method using trigonometric identities. Graphically and analytically, it is shown that these results differ only by a constant.

Step-by-step solution

Method 1 - Using Substitution

Let \( u = \sec 3x \), then \( du = 3 \sec 3x \tan 3x dx \). Moreover, the differential equation becomes \(\int u^3 du\). Solving this integral gives \( \frac{1}{4}u^4 + C\). After substituting \( u \) back, we get \( \frac{1}{4}(\sec^4 3x) + C \).

Method 2 - Using Trigonometric Identities

We can use the trigonometric identity \( \sec^2 x = 1 + \tan^2 x \) to rewrite the integral as \( \int (1 + \tan^2 3x)^2 \tan^3 3x dx \). After expanding and rearranging, we now need to find the integrals to \( \int \tan^3 3x dx \(, \( \int \tan^5 3x dx \), and \( \int \tan^7 3x dx \). Solving these integrals provides us the antiderivative.

Graphing the Antiderivatives

When graphing the antiderivative obtained from both methods, we can see that the curves overlap and are identical except for a vertical shift. This shift equals to the constant of integration.

Verifying Analytically

Select a value for \( x \) and calculate the value of both functions obtained from method 1 and method 2. Subtract these two values, and you will see that the difference is a constant, which verifies our work analytically.

Key concepts

Integration by Substitution

When tackling an integral that is difficult to solve directly, integration by substitution can be a powerful tool. This technique is similar to the chain rule in differentiation. To use it, you begin by identifying a section of the integral that you can set as a new variable, usually denoted as \( u \). When \( u \) is a function of \( x \), you find \( \frac{du}{dx} \), and from there, you can express \( dx \) in terms of \( du \). Once the substitution is made, the integral often becomes easier to solve. After integrating with respect to \( u \), you substitute back in terms of \( x \) to get the final solution.

A good example can be seen in the exercise, where the substitution \( u = \sec 3x \) simplifies the original integral to \( \int u^3 du \), which is much more straightforward. Remember to adjust your differential accordingly; as shown in the steps, \( du = 3 \sec 3x \tan 3x dx \) means we need to account for that '3' in the integral as well. The final step is always to revert back to the original variable to get your antiderivative in terms of \( x \).

Trigonometric Identities

The realm of integration often involves trigonometric functions, where trigonometric identities become invaluable. These identities allow for the simplification of trigonometric expressions which can make integrals more tractable. Common identities include \( \sin^2 x + \cos^2 x = 1 \), \( 1 + \tan^2 x = \sec^2 x \), and various angle sum and difference formulas.

For instance, in our exercise, one approach is to leverage the identity \( \sec^2 x = 1 + \tan^2 x \) to transform \( \sec^4 3x \tan^3 3x \) into a polynomial in terms of \( \tan 3x \). Then, you're left with simpler integrals like \( \int \tan^3 3x dx \), which can be approached using traditional integration methods. Using these identities effectively requires practice, but they are pivotal in simplifying complex trigonometric integrals.

Graphing Antiderivatives

Visualizing math can provide a deeper understanding, and this is where graphing antiderivatives comes in. Every function has its own antiderivative, but when you calculate the indefinite integral, you obtain a family of functions because of the constant of integration \( C \).

To graph antiderivatives, you can use software or a graphing utility. When doing so, it's common to omit the constant \( C \) for simplicity's sake. After graphing, you should observe that antiderivatives that stem from the same function differ only by vertical shifts. This was evident in the exercise, as graphing the antiderivatives obtained by two different methods resulted in the same curve shape, only shifted by the constant \( C \). Understanding how to graph antiderivatives not only aids in visual comprehension but also in verifying the correctness of your integration.

Analytical Verification

In mathematics, it's crucial to validate your solutions, which brings us to analytical verification. This process establishes that different methods of solving a calculus problem yield consistent results. Once you've reached an antiderivative, you can perform analytical verification by simply plugging in values for the variable and comparing the outcomes. If two antiderivatives differ only by a constant, their difference should yield that constant regardless of the input value.

Returning to our original exercise, by choosing a value for \( x \) and computing the antiderivative with both methods, the subtraction of one from the other should indeed reveal the constant of integration. This sort of verification is not only a critical tool in confirming the accuracy of your solutions but also in reinforcing your understanding of the underlying mathematics of integral calculus.