Question

Evaluate the indefinite integral: \[ \int e^x (\sin x + \cos x)\, dx \]

• 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:

Whenever you see \(e^x\) multiplied by a sum of two functions, check if one is the derivative of the other. It often indicates a direct \(e^x[f(x)+f'(x)]\) pattern.

Answer 1

The Correct Option is B

Concept: This integral is based on recognizing a standard pattern: \[ \int e^x [f(x) + f'(x)]\,dx = e^x f(x) + C \] This comes directly from the product rule: \[ \frac{d}{dx}(e^x f(x)) = e^x f(x) + e^x f'(x) \]

Step 1: Identify the pattern.
Given: \[ I = \int e^x (\sin x + \cos x)\,dx \] Compare with \(f(x) + f'(x)\): Let \[ f(x) = \sin x \quad \Rightarrow \quad f'(x) = \cos x \] So the integrand becomes: \[ e^x [f(x) + f'(x)] \]

Step 2: Apply the formula.
Using: \[ \int e^x [f(x) + f'(x)]\,dx = e^x f(x) + C \] We get: \[ I = e^x \sin x + C \]

Step 3: Verification (optional idea).
Differentiate \(e^x \sin x\): \[ \frac{d}{dx}(e^x \sin x) = e^x \sin x + e^x \cos x = e^x(\sin x + \cos x) \] This matches the integrand, so the result is correct.