Definite Integration
IBP Repeated
Grade None
Question:
<p>Evaluate \(\displaystyle\int_0^1 x^2 e^x\,dx\) [JEE Main 2018]</p>
Step-by-Step Solution
Key Concept: IBP twice: \intx^2eˣdx = x^2eˣ - 2xeˣ + 2eˣ + C = eˣ(x^2-2x+2)+C.
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<p>$\int x^2e^x dx = x^2e^x-2\int xe^x dx = x^2e^x-2(xe^x-e^x)+C = e^x(x^2-2x+2)+C$</p>
<p>$$\int_0^1 x^2e^x dx = [e^x(x^2-2x+2)]_0^1 = e(1-2+2)-e^0(0-0+2)=e-2$$</p>
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Correct Answer: A