Question

The value of \( \int e^{x(\sin x + \cos x)} \, dx \) is:

• A. $e^{x} \cos x + c$
• B. $e^{x} \sin x + c$
• C. $-e^{x} \sin x + c$
• D. $e^{x} (\sin x - \cos x) + c$

Hint:

Recognizing this $e^x$ pattern can save you from a lengthy integration by parts.

Answer 1

The Correct Option is B

Step 1: Concept
Use the standard integral form: $\int e^{x}[f(x) + f'(x)] dx = e^{x}f(x) + c$.
Step 2: Analysis
Let $f(x) = \sin x$. Then $f'(x) = \cos x$. The integrand matches the form $e^{x}[f(x) + f'(x)]$.
Step 3: Conclusion
The integral is $e^{x} \sin x + c$.
Final Answer: (B)