Integral-Calculus Question 515
Question: What is the value of $ \int_0^{1}{x{e^{x^{2}}}dx} $ ?
Options:
A) $ \frac{(e-1)}{2} $
B) $ e^{2}-1 $
C) $ 2(e-1) $
D) $ e-1 $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Let $ I=\int\limits_0^{1}{x{e^{x^{2}dx}}} $ Let $ x^{2}=t $
$ \Rightarrow 2xdx=dt $
$ \Rightarrow xdx=\frac{dt}{2} $ when $ x=0,t=0 $ then $ x=1,t=1 $ $ x=1,t=1 $
$ \Rightarrow I=\frac{1}{2}\int\limits_0^{1}{e^{t}dt}=\frac{1}{2}[ e^{t} ]_0^{1} $ $ =\frac{1}{2}[ {e^{x^{2}}} ]_0^{1}=\frac{1}{2}[e-e^{0}]=\frac{e-1}{2} $
BETA
