Question

Evaluate \(\int e^{x}(\sin x+2\,\cos x)\sin xdx\)

• A. \(e^{x}\cos x+C\)
• B. \(e^{x}\sin x+C\)
• C. \(e^{x}\sin ^{2}x+C\)
• D. \(e^{x}\sin 2x+C\)

Hint:

Expand the integrand and recognise it as the derivative of e^x times sin squared x.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Expand: \((\sin x+2\cos x)\sin x=\sin^{2}x+2\sin x\cos x\).

Step 2: Recognise the Form
\(2\sin x\cos x=\dfrac{d}{dx}(\sin^{2}x)\). So the integrand is \(e^{x}\left[f(x)+f'(x)\right]\) with \(f(x)=\sin^{2}x\).

Step 3: Apply the Standard Result
\(\int e^{x}[f(x)+f'(x)]dx=e^{x}f(x)+C\).

Step 4: Result
The integral is \(e^{x}\sin^{2}x+C\).

Final Answer:
The integral equals \(e^{x}\sin^{2}x+C\). \[ \boxed{\text{(C) }e^{x}\sin ^{2}x+C} \]