Question

\( \int \left(\tan^{2}(2x) - \cot^{2}(2x)\right)\,dx \)

• A. \(\frac{-1}{2} (\tan 2x + \cot 2x) + C\)
• B. \(2 (\tan 2x + \cot 2x) + C\)
• C. \(\frac{1}{2} (\tan 2x - \cot 2x) + C\)
• D. \(\frac{-1}{2} (\tan 2x - \cot 2x) + C\)
• E. \(\frac{1}{2} (\tan 2x + \cot 2x) + C\)

Hint:

When integrating \(\tan^n x\) or \(\cot^n x\), always look to use the Pythagorean identities first. It reduces the degree and brings the expression into a form where the derivative of the base function is present.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The integral involves the squares of the tangent and cotangent functions.
Since there are no direct, simple standard formulas for integrating \( \tan^2(kx) \) or \( \cot^2(kx) \) directly, we must transform the integrand using fundamental trigonometric identities.
By converting these into secant and cosecant squared functions, the integration becomes straightforward because they are standard derivatives.
Step 2: Key Formula or Approach:
We will utilize the fundamental Pythagorean trigonometric identities:
1. \( \tan^2 \theta = \sec^2 \theta - 1 \)
2. \( \cot^2 \theta = \csc^2 \theta - 1 \)
The standard integral formulas required are:
1. \( \int \sec^2(kx) dx = \frac{1}{k}\tan(kx) + C \)
2. \( \int \csc^2(kx) dx = -\frac{1}{k}\cot(kx) + C \)
Step 3: Detailed Explanation:
Let the given integral be \( I \).
\[ I = \int (\tan^2(2x) - \cot^2(2x)) dx \] Apply the trigonometric identities to replace \( \tan^2(2x) \) and \( \cot^2(2x) \):
\[ I = \int \left( (\sec^2(2x) - 1) - (\csc^2(2x) - 1) \right) dx \] Distribute the negative sign and simplify the terms inside the integral:
\[ I = \int (\sec^2(2x) - 1 - \csc^2(2x) + 1) dx \] The \( -1 \) and \( +1 \) cancel each other out, leaving:
\[ I = \int (\sec^2(2x) - \csc^2(2x)) dx \] Now, use the linearity property of integrals to split it into two separate integrals:
\[ I = \int \sec^2(2x) dx - \int \csc^2(2x) dx \] Apply the standard integration formulas as stated in Step 2, noting that our coefficient \( k \) is 2:
\[ I = \left( \frac{1}{2}\tan(2x) \right) - \left( -\frac{1}{2}\cot(2x) \right) + C \] Simplify the signs:
\[ I = \frac{1}{2}\tan(2x) + \frac{1}{2}\cot(2x) + C \] Factor out the common \( \frac{1}{2} \):
\[ I = \frac{1}{2}(\tan 2x + \cot 2x) + C \] Step 4: Final Answer:
The integrated expression is \( \frac{1}{2}(\tan 2x + \cot 2x) + C \).