\(\int(2x+1)e^{2x}\,dx\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(xe^{2x}+C\)
Using integration by parts with \(2x+1\) as the first function and \(e^{2x}\) as the second (\(\int e^{2x}\,dx = \frac{1}{2}e^{2x}\)):
\( \displaystyle \int(2x+1)e^{2x}\,dx = \frac{1}{2}(2x+1)e^{2x}-\int 2\cdot\frac{1}{2}e^{2x}\,dx \)
\( \displaystyle = \frac{1}{2}(2x+1)e^{2x}-\frac{1}{2}e^{2x}+C \)
\( = xe^{2x}+C \)
Check: \(\frac{d}{dx}\left(xe^{2x}\right) = e^{2x}+2xe^{2x} = (2x+1)e^{2x}\).
Why the other options are incorrect
Option (2) \((x+1)e^{2x}+C\) — Incorrect. Its derivative is \((2x+3)e^{2x}\).
Option (3) \(\displaystyle\frac{1}{2}(2x+1)e^{2x}+C\) — Incorrect. It stops after the first term of the by-parts formula and leaves out \(-\frac{1}{2}e^{2x}\).
Option (4) \(\left(x^2+x\right)e^{2x}+C\) — Incorrect. It integrates \(2x+1\) and keeps \(e^{2x}\) unchanged; its derivative is \((2x^2+4x+1)e^{2x}\).