Question

$\int (x+1)^{2} e^{x} dx =$

• A. $x e^{x} + c$
• B. $x^{2} e^{x} + c$
• C. $(x+1) e^{x} + c$
• D. $(x^{2}+1) e^{x} + c$

Hint:

Whenever you integrate $e^{x}$ multiplied by a polynomial, try to arrange it as $f(x) + f'(x)$.

Answer 1

The Correct Option is D

Step 1: Concept
Expand the square and use the standard integral form $\int e^{x}[f(x) + f'(x)] dx = e^{x} f(x) + c$.

Step 2: Meaning
$\int (x^{2} + 2x + 1) e^{x} dx = \int e^{x} [x^{2} + 1 + 2x] dx$.

Step 3: Analysis
Let $f(x) = x^{2} + 1$. Then $f'(x) = 2x$. The integral is of the form $e^{x}[f(x) + f'(x)]$.

Step 4: Conclusion
The result is $e^{x} f(x) = e^{x}(x^{2} + 1) + c$. Final Answer: (D)