Question

\( \displaystyle \int \ cot x (1-\ cosec x)e^x\,dx \) is

• A. \( e^x\ cot x + C \)
• B. \( -e^x\ cot x + C \)
• C. \( e^x\ cosec x + C \)
• D. \( -e^x\ cosec x + C \)
• E. \( e^x\ cosec x\ cot x + C \)

Hint:

For integrals involving trigonometric functions like \( \ cot x \) and \( \ cosec x \), integration by parts can be a useful method. Always carefully separate the terms to simplify the process.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The problem asks for the indefinite integral of \( e^x \cot x(1 - \csc x) \). The presence of \( e^x \) multiplied by a trigonometric function often suggests using the integration formula \( \int e^x (f(x) + f'(x)) dx = e^x f(x) + C \).
Step 2: Key Formula or Approach:
Let's expand the integrand and see if it fits the form \( e^x (f(x) + f'(x)) \).
\[ \int e^x (\cot x - \cot x \csc x) dx \] Let's try different choices for \( f(x) \).
Case 1: Let \( f(x) = \cot x \). Then \( f'(x) = -\csc^2 x \). The integrand is \( e^x(\cot x - \cot x \csc x) \), which is not of the form \( e^x(f(x)+f'(x)) \).
Case 2: Let \( f(x) = -\csc x \). Then \( f'(x) = -(-\csc x \cot x) = \csc x \cot x \). The expression \( f(x) + f'(x) = -\csc x + \csc x \cot x \). This also does not match the expression in the integral.
Case 3: Let \( f(x) = -\cot x \csc x \). This function's derivative is more complex and unlikely to lead to a simple solution.
Step 3: Detailed Explanation:
The integrand \( e^x (\cot x - \cot x \csc x) \) does not simplify into the standard form \( e^x (f(x) + f'(x)) \) with a simple choice of \( f(x) \). Attempting to solve this integral using integration by parts would be very complicated and would not lead to any of the simple options provided. This suggests there is likely a typo in the question itself. For instance, if the question was \( \int e^x(\cot x - \csc^2 x) dx \), we could set \( f(x) = \cot x \) and \( f'(x) = -\csc^2 x \), and the answer would be \( e^x \cot x + C \). Given that none of the standard methods lead to the given options, the question is considered flawed.
Step 4: Final Answer:
As the problem does not conform to standard integration patterns and cannot be solved to match any of the given options, the question is invalid. In many competitive exams, such questions are cancelled.