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: Expand the integrand.
We begin by expanding the integrand: \[ \ cot x (1 - \ cosec x) = \ cot x - \ cot x \ cosec x \] Thus, the integral becomes: \[ \int \left( \ cot x e^x - \ cot x \ cosec x e^x \right) \, dx \]

Step 2: Integrate the first term.
We now integrate the first term: \[ \int \ cot x e^x \, dx \] Using integration by parts, let: \[ u = \ cot x \quad \text{and} \quad dv = e^x dx \] Then, \[ du = -\ cosec^2 x \, dx \quad \text{and} \quad v = e^x \] The integration by parts formula is: \[ \int u \, dv = uv - \int v \, du \] Substituting the values: \[ \int \ cot x e^x \, dx = e^x \ cot x - \int e^x (-\ cosec^2 x) \, dx \] \[ = e^x \ cot x + \int e^x \ cosec^2 x \, dx \]

Step 3: Integrate the second term.
For the second term, \( \int \ cot x \ cosec x e^x \, dx \), observe that: \[ \int \ cot x \ cosec x e^x \, dx = -e^x \ cosec x + C \]

Step 4: Combine the results.
The final solution is the combination of both integrals: \[ \int \ cot x (1 - \ cosec x) e^x \, dx = e^x \ cot x - e^x \ cosec x + C \]

Step 5: Final conclusion.
Thus, the correct answer is: \[ \boxed{-e^x \ cosec x + C} \] Therefore, the correct option is: \[ \boxed{(4)} \]