Question

Solve the integral \(\int \frac{\sec^2 2\theta}{(\cot \theta - \tan \theta)^2} \, d\theta\).

Hint:

When faced with a complex trigonometric integral, the first step should always be to simplify the integrand as much as possible using identities. Look for expressions that can be simplified into double angles (like \(\cot \theta - \tan \theta\)) or that are derivatives of other parts of the expression, suggesting a substitution.

Answer 1

Step 1: Understanding the Concept:
This problem involves solving an indefinite integral containing trigonometric functions. The key to solving it is to simplify the integrand using trigonometric identities before performing the integration, which will likely involve a substitution.
Step 2: Key Formula or Approach:
The solution uses the following key trigonometric identities: 1. \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) 2. \(\cos^2 \theta - \sin^2 \theta = \cos 2\theta\) 3. \(2\sin \theta \cos \theta = \sin 2\theta\) 4. \(\frac{\cos 2\theta}{\sin 2\theta} = \cot 2\theta\) 5. \(\frac{d}{d\theta}(\tan 2\theta) = 2\sec^2 2\theta\) The integration will be performed using the method of substitution.
Step 3: Detailed Explanation or Calculation:
Let the integral be \(I\). First, simplify the denominator: \[ \cot \theta - \tan \theta = \frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta} = \frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta} \] Using the double angle identities, \(\cos^2 \theta - \sin^2 \theta = \cos 2\theta\) and \(\sin \theta \cos \theta = \frac{1}{2}\sin 2\theta\): \[ \cot \theta - \tan \theta = \frac{\cos 2\theta}{\frac{1}{2}\sin 2\theta} = 2 \cot 2\theta \] Now substitute this back into the integral: \[ I = \int \frac{\sec^2 2\theta}{(2\cot 2\theta)^2} d\theta = \int \frac{\sec^2 2\theta}{4\cot^2 2\theta} d\theta \] Rewrite the integrand in terms of \(\tan\) and \(\sec\). Since \(\cot 2\theta = \frac{1}{\tan 2\theta}\): \[ I = \frac{1}{4} \int \frac{\sec^2 2\theta}{1/\tan^2 2\theta} d\theta = \frac{1}{4} \int \tan^2 2\theta \sec^2 2\theta d\theta \] Now, we use the substitution method. Let \(u = \tan 2\theta\). Then, the derivative is \(du = \frac{d}{d\theta}(\tan 2\theta) d\theta = 2\sec^2 2\theta d\theta\).
This gives us \(\sec^2 2\theta d\theta = \frac{1}{2} du\).
Substitute \(u\) and \(du\) into the integral: \[ I = \frac{1}{4} \int u^2 \left(\frac{1}{2} du\right) = \frac{1}{8} \int u^2 du \] Now, integrate with respect to \(u\): \[ I = \frac{1}{8} \left(\frac{u^3}{3}\right) + C = \frac{u^3}{24} + C \] Finally, substitute back \(u = \tan 2\theta\): \[ I = \frac{\tan^3 2\theta}{24} + C \] Step 4: Final Answer:
The solution to the integral is \(\frac{\tan^3 2\theta}{24} + C\).