Definite Integration
Definite + Exponential-Rational
Grade 12
Question:
<p>Evaluate \(\displaystyle\int_{-1}^1 xe^{|x|}\,dx\) [JEE Main 2022]</p>
<li>\(0\)</li>
<li>\(2(e-1)\)</li>
<li>\(e-1\)</li>
<li>\(2e\)</li>
Step-by-Step Solution
Key Concept: xe^|x| is an odd function (odd \times even = odd). Integral over symmetric interval [-1,1] = 0.
<div class='solution'>
<p>Let $f(x)=xe^{|x|}$. Then $f(-x)=(-x)e^{|-x|}=-xe^{|x|}=-f(x)$: <strong>odd function</strong>.</p>
<p>$$\int_{-1}^1 xe^{|x|}dx = \boxed{0}$$</p>
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Correct Answer: A