Question

Perform the indicated integrations. $$ \int \cot ^{5} 2 t d t $$

Short answer

\[ \frac{1}{8} (\sin 2t)^{-4} + (\sin 2t)^{-2} - \frac{1}{2} \ln |\sin 2t| + C \]

Step-by-step solution

Rewrite in terms of basic trigonometric identities

Start by expressing the integral in terms of sine and cosine functions. Recall that \( \cot x = \frac{\cos x}{\sin x} \). The integral becomes:\[ \int (\cot 2t)^5 \, dt = \int \left( \frac{\cos 2t}{\sin 2t} \right)^5 \, dt = \int \frac{\cos^5 2t}{\sin^5 2t} \, dt \]

Use substitution to simplify the integral

Let \( u = \sin 2t \), then \( \frac{du}{dt} = 2 \cos 2t \) or \( dt = \frac{du}{2 \cos 2t} \). Substitute \( u \) and \( dt \) in the integral:\[ \int \frac{\cos^5 2t}{u^5} \, \frac{du}{2 \cos 2t} = \frac{1}{2} \int \frac{\cos^4 2t}{u^5} \, du \]

Simplify using trigonometric identities

Use the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \), so \( \cos^2 2t = 1 - u^2 \) and in the fourth power: \( \cos^4 2t = (1 - u^2)^2 \). Substitute back into the integral:\[ \frac{1}{2} \int \frac{(1 - u^2)^2}{u^5} \, du \]

Expand the expression in the integral

Expand \((1 - u^2)^2\) resulting in: \[ \frac{1}{2} \int \frac{1 - 2u^2 + u^4}{u^5} \, du = \frac{1}{2} \int \left( u^{-5} - 2u^{-3} + u^{-1} \right) \, du \]

Integrate each term separately

Perform the integration of each term separately: 1. \( \int u^{-5} \, du = \frac{-1}{4} u^{-4} + C_1 \)2. \( \int u^{-3} \, du = \frac{-1}{2} u^{-2} + C_2 \)3. \( \int u^{-1} \, du = \ln |u| + C_3 \)Combine these results:\[ \frac{1}{2} \left( \frac{-1}{4} u^{-4} - 2\left(\frac{-1}{2} u^{-2}\right) + \ln |u| \right) = \frac{1}{8} u^{-4} + u^{-2} - \frac{1}{2} \ln |u| + C \]

Substitute back the original variable

As \( u = \sin 2t \), substitute \( u \rightarrow \sin 2t \):\[ \frac{1}{8} (\sin 2t)^{-4} + (\sin 2t)^{-2} - \frac{1}{2} \ln |\sin 2t| + C \]

Key concepts

Trigonometric Integration

When solving integrals involving trigonometric functions, it's often necessary to rewrite the expressions using trigonometric identities. In our exercise, the cotangent function \( \cot x = \frac{\cos x}{\sin x} \) is expressed in terms of sine and cosine forms.
This transformation simplifies the integration process, making it easier to handle complex trigonometric integrals.

Applying trigonometric identities is crucial because:

• They simplify the functions by converting them into forms that are easier to integrate.
• They allow the use of simple calculus techniques on complex expressions.

In our example, by expressing \( \cot 2t \) as \( \frac{\cos 2t}{\sin 2t} \), we prepare the integrand for further simplifications through substitution and identity usage.
Remember, understanding these trigonometric form transformations is a key skill in integral calculus.

Substitution Method

The substitution method in calculus is a powerful tool that replaces a variable in an integral with another variable, essentially simplifying the equation to make it easier to solve. In our integral, we use substitution to tackle the complex form of \( \cot^{5} 2t \).
This method involves several steps:

• Choosing a new variable to replace a complex part of the integral, such as letting \( u = \sin 2t \).
• Finding \( \frac{du}{dt} \) to express \( dt \) in terms of the new variable.
• Substituting these expressions into the integral to simplify it.

The replacement results in a more straightforward expression, here allowing us to move from a trigonometric function to a polynomial one.
Through substitution, integration becomes manageable, transforming a potentially complicated integral into multiple simpler integrals that can be addressed individually.
This technique is especially beneficial in trigonometric integration, where direct integration is often challenging without simplification.

Pythagorean Identity

The Pythagorean identity is a foundational concept in trigonometry used to rewrite expressions involving squared trigonometric functions. The identity \( \cos^2 x = 1 - \sin^2 x \) is particularly helpful when simplifying integrands.
In our problem, once we've substituted \( u = \sin 2t \), we apply the Pythagorean identity to \( \cos^2 2t \) converting it to \( 1 - u^2 \).
This transformation is critical due to:

• Reducing the power of cosine functions in integrals, simplifying polynomial expressions.
• Allowing further substitution or expansion processes to simplify the integration further.

In the task, \( \cos^4 2t \) is simplified to \( (1 - u^2)^2 \), resulting in expressions that are easier to integrate term by term.
Mastering this and similar identities not only aids in solving trigonometric integrals but also enhances overall problem-solving skills in calculus.