Question

\(\int (x+1)e^x\,dx\) is equal to
[ \(C\) is an arbitrary constant]

• A. \((x+1)e^x+C\)
• B. \(e^x+C\)
• C. \(xe^x+C\)
• D. \((x+1)+C\)

Hint:

Use \(\int e^x(f+f')dx=e^xf\) with \(f=x\), or integrate by parts.

Answer 1

The Correct Option is C

Step 1: Split the integrand.
$(x+1)e^x = xe^x+e^x$, so the integral splits into two simple parts.

Step 2: Use a known standard form.
We know $\int e^x(f(x)+f'(x))\,dx=e^xf(x)+C$. Take $f(x)=x$. Then $f'(x)=1$, and $e^x(x+1)$ is exactly $e^x(f+f')$.

Step 3: Write the answer.
\[ \int e^x(x+1)\,dx = x e^x + C \]

Step 4: Check by differentiation.
$\frac{d}{dx}(xe^x)=e^x+xe^x=(x+1)e^x$. This gives back the integrand, so the answer is right.

Final Answer:
So the result is $xe^x+C$. \[ \boxed{xe^x+C} \]