Problem 46 - Essential Calculus

Question

Find the integral. Use a computer algebra system to confirm your result. $$ \int \csc ^{2} 3 x \cot 3 x d x $$

Step-by-Step Solution

Answer

The solution to the integral \(\int \csc^{2}3x \cdot \cot3x dx\) is \(-\frac{1}{3\sin3x}\).

Step 1: Recognition and Power Reducing

Recognize that the integral \(\int \csc^{2}3x \cdot \cot3x dx\) can be rewritten in terms of sine and cosine as \(\int \frac{\cos3x}{\sin^{2}3x} dx\) to make it simpler for further steps.

Step 2: Apply the Substitution Method

Initiate a substitution where \(u = \sin3x\), so that \(du = 3\cos3x dx\). The original integral can thus be rewritten as \(\frac{1}{3} \int \frac{1}{u^{2}} du\).

Step 3: Integrate and Substitute Back

The integral \(\frac{1}{3} \int \frac{1}{u^{2}} du\) is relatively simple and evaluates to \(-\frac{1}{3u}\) . Substituting back \(u = \sin3x\) gives us the solution to the original problem as \(-\frac{1}{3\sin3x}\).

Key Concepts

Substitution Method

The substitution method in integral calculus is a key technique used to simplify the process of finding integrals. It's akin to a change of variable that makes an integral more manageable. To effectively use the substitution method, one typically looks for a function inside the integral that, when substituted with a simpler variable (say, u), simplifies the integrand. For instance, in the integral
\[ \frac{1}{3} \int \frac{1}{u^{2}} du \]
we arrived here by setting u as sin(3x), making du equal to 3cos(3x) dx. This shift not only makes the integral less complex but also allows us to solve it without too much difficulty. It's critical to also adjust the differential (in this case, dx to du) to match the new variable. Once you integrate using u, you'll need to substitute back to get the answer in terms of the original variable.

Power Reducing

Power reducing is another technique often used in integration, especially when you're confronted with trigonometric functions raised to a power. It involves using identities that re-express powers of sine and cosine in forms that are easier to integrate.

In the example problem, the integral of cosecant squared times cotangent, these functions are powers and products of sines and cosines. These can typically be simplified by rewriting them using power-reducing formulas. However, in this case, recognizing that cotangent is the cosine over sine, and cosecant squared is 1/sine squared, directly simplifies the integral without a separate power-reducing step. This simplicity is what makes recognizing and applying trigonometric identities an important skill in solving integrals.

Cotangent Integration

Integrating the cotangent function, \( \cot x \), may initially seem daunting due to its less frequent appearance compared to its trigonometric counterparts. However, through some simplification using trigonometric identities, cotangent integration becomes quite approachable.

Recall that cotangent is actually the reciprocal of the tangent function and can also be written as the ratio of the cosine to the sine function (cos(x)/sin(x)). This relationship is used to rewrite the integral in terms that may reveal a simpler path to the solution. In the problem we are considering, \( \int \csc^{2}3x \cdot \cot3x dx \), the presence of cotangent is pivotal; it sets up the substitution that simplifies the integral significantly. Remember, by turning unfamiliar and complex expressions into familiar forms, we can unravel challenging integrals with confidence.

Cosecant Integration

The integration of cosecant, denoted as \( \csc x \), is often considered arduous because its occurrences in standard integration problems are less frequent. To integrate functions involving cosecant, one must become comfortable working with its definition as the reciprocal of sine, 1/sin(x), and its related identities.

In our example, cosecant squared appears which is csc^2(x) or 1/sin^2(x). This makes it a prime candidate for substitution when combined with functions like cotangent. After the substitution, as shown in our provided problem, the process of integration becomes a matter of a simple power integration. Thus, understanding how cosecant behaves, its relationship to sine, and when to pair it with other trigonometric functions, significantly aids in solving integrals that at first, may not seem directly integrable.