Question

\(\displaystyle \int e^{x}\,(x^{3}+3x^{2})\,dx=\) ?

• A. \(3x^{2}e^{x}+k\)
• B. \(x^{2}e^{x}+k\)
• C. \(x^{3}e^{x}+k\)
• D. \(3e^{x}\cdot x^{3}+k\)

Hint:

When you see $e^x$ times a polynomial, try derivative of $(\text{polynomial})\cdot e^x$.

Answer 1

The Correct Option is C

Check the derivative of \(x^{3}e^{x}\) by product rule: \[ \frac{d}{dx}(x^{3}e^{x})=e^{x}x^{3}+e^{x}\cdot 3x^{2}=e^{x}(x^{3}+3x^{2}). \] It matches the integrand, so the antiderivative is \(x^{3}e^{x}+k\).