Problem 32 - Calculus: One and Several Variables

Question

Calculate. (If you run out of ideas, use the examples as models.) $$\int \cot ^{5} 2 x d x$$.

Step-by-Step Solution

Answer

The short version of the answer is: \[ \int \cot ^{5} 2 x d x = -\frac{1}{8}(\sin 2x)^{-4} + C \]

Step 1: Rewrite the cotangent function in terms of sine and cosine

Recall that the cotangent function is the reciprocal of the tangent function, which can also be expressed in terms of sine and cosine functions as follows: \[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \] So, our given integral becomes: \[ \int (\frac{\cos 2x}{\sin 2x})^5 dx \]

Step 2: Perform substitution

Let the substitution be \(u = \sin 2x\). Now, find the derivative of u with respect to x: \[ \frac{du}{dx} = \frac{d}{dx}(\sin 2x) = 2\cos 2x \] So, \[ dx = \frac{du}{2\cos 2x} \] Replace dx in the integral, and simplify the expression as follows: \[ \int (\frac{\cos 2x}{\sin 2x})^5 dx = \int (\frac{1}{u})^5 \frac{du}{2\cos 2x} \] Since \(\cot 2x = \frac{\cos 2x}{\sin 2x}\), we can rewrite the integral as follows: \[ \int (\frac{1}{u})^5 \frac{du}{2\cot 2x \cdot u} = \frac{1}{2}\int u^{-5} du \]

Step 3: Integrate u

Now integrate with respect to u: \[ \frac{1}{2}\int u^{-5} du = \frac{1}{2} \cdot \frac{u^{-4}}{-4} + C = -\frac{1}{8}u^{-4} + C \]

Step 4: Substitute back the original function

Finally, substitute \(u = \sin 2x\) back into the equation: \[ -\frac{1}{8}u^{-4} + C = -\frac{1}{8}(\sin 2x)^{-4} + C \] So, the final answer is: \[ \int \cot ^{5} 2 x d x = -\frac{1}{8}(\sin 2x)^{-4} + C \]

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a valuable technique in calculus for simplifying integrals that involve trigonometric functions. By using a substitution method, we can often convert complex trigonometric expressions into rational functions that are easier to integrate.

Here, in the exercise, we employed a substitution by setting the variable:

• \( u = \sin 2x \)

When we do this, it transforms the original integral involving variables in terms of \( x \) into an integral in terms of \( u \). This makes the integration process more manageable.

We avoid the trigonometric complexity in the original integral by rewriting it in a simpler form, using:

• \( \cot 2x = \frac{\cos 2x}{\sin 2x} \)

After substituting \( dx \) and modifying the function accordingly, our integral problem becomes straightforward. The main idea here is to make the integral more approachable, and substitution often paves the way for integration when faced with trigonometric functions.

Remember, when substituting in integrals: always replace all instances of the variable you are substituting for, including the differential, like \( dx \). This will ensure you have a proper integral with respect to your new variable throughout.

Definite and Indefinite Integrals

Integrals can be categorized into two types: definite and indefinite integrals. The integral in our exercise is an example of an indefinite integral, as it lacks specific limits. Indefinite integrals represent a family of functions and include a constant \( C \), known as the constant of integration, due to the antiderivative's arbitrary nature.

In our exercise:

• The integration process begins with rewriting the function for ease.
• We then perform the integration concerning the new variable.
• Finally, we substitute back our original terms to express the integral in terms of the initial variable.

Indefinite integrals are crucial as they form the basis for computing area and solving differential equations. They provide a general solution or family of solutions.

It's important to understand that while definite integrals give specific values representing areas under curves between set limits, indefinite integrals focus on broader function analysis, showing possible accumulated values without bound limitations.

Trigonometric Identities

Trigonometric identities are crucial tools in simplifying and solving integrals involving trigonometric functions. They represent equations involving trigonometric functions that hold true for all values of the variable where both sides of the equation are defined.

In the exercise, we utilized the identity:

• \( \cot x = \frac{\cos x}{\sin x} \)

This identity helped transform the original integral into a more workable form by expressing the cotangent function, which simplifies the integration process.

Trigonometric identities can transform complex integrals into simpler ones, allowing us to perform integrations that might otherwise be challenging.

Commonly used identities include the Pythagorean identities, angle sum and difference identities, and double angle identities. Mastering these can significantly ease the process of solving integrals, particularly in calculus problems involving trigonometric functions.

Importantly, using these identities can help identify patterns and simplify expressions early on, which makes calculating the integral or derivative much more straightforward.