Question

For the following exercises, find the definite or indefinite integral. $$ \int \cot (3 x) d x $$

Short answer

\( \int \cot(3x) \, dx = \frac{1}{3} \ln |\sin(3x)| + C \)

Step-by-step solution

Rewrite Cotangent in Terms of Sine and Cosine

Recall that the cotangent function can be rewritten in terms of sine and cosine as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). Substitute in the integral to get: \[ \int \cot(3x) \, dx = \int \frac{\cos(3x)}{\sin(3x)} \, dx \]

Use a Substitution

To solve this integral, use the substitution method. Let \( u = \sin(3x) \). Then, the derivative \( du = 3 \cos(3x) \, dx \) implies that \( dx = \frac{1}{3 \cos(3x)} \, du \). Substitute \( u \) and \( dx \) in the integral:\[ \int \frac{\cos(3x)}{u} \cdot \frac{1}{3 \cos(3x)} \, du = \frac{1}{3} \int \frac{1}{u} \, du \]

Integrate Using the Natural Logarithm

The integral \( \int \frac{1}{u} \, du \) is a standard integral that equals \( \ln |u| + C \), where \( C \) is the constant of integration. Therefore, our integral becomes:\[ \frac{1}{3} \ln |u| + C \]

Substitute Back

Replace \( u \) with \( \sin(3x) \) to get the solution in terms of \( x \). The final expression is:\[ \frac{1}{3} \ln |\sin(3x)| + C \]

Final Answer

The indefinite integral \( \int \cot(3x) \, dx \) is thus:\[ \frac{1}{3} \ln |\sin(3x)| + C \]

Key concepts

Indefinite Integrals

Indefinite integrals are a cornerstone of calculus, often referred to as finding an antiderivative. But what do we mean by this? Simply put, an indefinite integral is the reverse process of differentiation. Instead of finding the rate of change of a function, we are figuring out how to recreate an original function from its derivative. The notation looks like this: \( \int f(x) \, dx \). In mathematical terms, the result is a family of functions, expressed as \( F(x) + C \), where \( C \) is a constant. This constant symbolizes the infinite number of vertical shifts an antiderivative can have on a graph.

Essentially, an indefinite integral does not have specified limits. It's more general and does not provide a numeric result. Its purpose is to find all possible functions whose derivative is the integrand, hence the presence of the constant \( C \).

Trigonometric Integrals

Trigonometric integrals involve functions such as \( \sin, \cos, \tan, \text{and} \cot \). These are integral calculations where trigonometric identities often come in handy. As these functions frequently appear in analysis involving circles and harmonic motions, they are an essential part of calculus.

The integration of trigonometric functions subscribes to identities and re-expressions to simplify complex expressions. In our given example, the integral \( \int \cot(3x) \, dx \) was re-expressed using the identity \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) to make the integration process more manageable.

By rewriting in terms of sine and cosine, it assists in employing substitution methods and simplifies the process of finding an antiderivative. This procedure is beneficial in resolving complicated integration scenarios that involve trigonometric expressions.

Substitution Method

The substitution method is akin to solving a puzzle. You are transforming the integral into a much simpler one by changing variables. Imagine it as swapping pieces of your equation to make everything fit together more easily.

Here's how it works: identify a part of the integral that can be replaced by a single variable \( u \). Then, replace \( dx \) with \( du \) by finding the derivative \( du = g'(x) \, dx \). This transforms the integral into one that is typically easier to solve.

In the provided example, the substitution was \( u = \sin(3x) \). Subsequently, \( du = 3 \cos(3x) \, dx \) allowed us to express the integral in terms of \( u \), changing it to \( \frac{1}{3} \int \frac{1}{u} \, du \). This step greatly simplifies the integration, leading to an integral that is straightforward to evaluate. Finally, one must substitute back to express the final solution in terms of the original variable.