Question:
\(\displaystyle \int e^{x}\,(x^{3}+3x^{2})\,dx=\) ?
\(\displaystyle \int e^{x}\,(x^{3}+3x^{2})\,dx=\) ?
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When you see $e^x$ times a polynomial, try derivative of $(\text{polynomial})\cdot e^x$.
Updated On: Oct 17, 2025
- \(3x^{2}e^{x}+k\)
- \(x^{2}e^{x}+k\)
- \(x^{3}e^{x}+k\)
- \(3e^{x}\cdot x^{3}+k\)
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The Correct Option is C
Solution and Explanation
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 \(x^{3}e^{x}+k\).
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