\(\int e^{2x}\left(2\sin x+\cos x\right)dx\) is equal to (C is a constant of integration)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(e^{2x}\sin x+C\)
By the product rule, for any function f(x):
\( \displaystyle \frac{d}{dx}\left[e^{2x}f(x)\right] = e^{2x}\left[2f(x)+f'(x)\right] \)
With \(f(x) = \sin x\), \(f'(x) = \cos x\), so the integrand is exactly this derivative of a product:
\( \displaystyle \frac{d}{dx}\left[e^{2x}\sin x\right] = e^{2x}\left(2\sin x+\cos x\right) \)
Hence
\( \int e^{2x}\left(2\sin x+\cos x\right)dx = e^{2x}\sin x+C \)
Why the other options are incorrect
Option (1) \(e^{2x}\cos x+C\) — Incorrect. Its derivative is \(e^{2x}\left(2\cos x-\sin x\right)\).
Option (2) \(\displaystyle\frac{1}{2}e^{2x}\sin x+C\) — Incorrect. Its derivative is half the integrand.
Option (3) \(2e^{2x}\sin x+C\) — Incorrect. Its derivative is twice the integrand.