$\displaystyle\int_0^{1} x e^{x^2}\,dx$ equals
A$\dfrac{1}{2}(e-1)$
B$e-1$
C$\dfrac{1}{2}e$
D$\dfrac{1}{2}(e+1)$
Answer & Solution
Correct answer: A. $\dfrac{1}{2}(e-1)$
1. Put $t=x^2$, so $dt=2x\,dx$, i.e. $x\,dx=\tfrac12 dt$.
2. Limits: $x=0\Rightarrow t=0$; $x=1\Rightarrow t=1$.
3. Integral $=\dfrac{1}{2}\int_0^1 e^{t}\,dt=\dfrac{1}{2}[e^{t}]_0^1=\dfrac{1}{2}(e-1)$.
4. Option B forgets the factor $\tfrac12$.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.337_
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