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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Evaluate the definite integral\[\int\limits_0^1x\;e^{x^2}dx\]

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Toolbox:
  • (i)$ \int \limits_a^bf(x)dx=F(b)-F(a)$
  • (ii)If there are two functions u and v, and the integral function is of the form $\int udv,$then it can be solved by the method of integration by parts.$ \int udv=uv-\int vdu$
  • (iii)$\int e^x=e^x+c.$
Given $\int\limits_0^1x\;e^{x^2}dx$
 
Let $ x^2=t$
 
On differentiating we get w.r.t.x
 
$2xdx=dt$
 
Therefore $ xdx=dt/2$
 
Substituting $t_1$ and dt we get
 
$I=\frac{1}{2}\int \limits_0^1 e^t dt$
 
On integrating we get,
 
$\frac{1}{2}[e^t]_0^1$
 
$=\frac{1}{2}[e^1-e^0]$
 
But $e^0=1$
 
$I=\frac{1}{2}[e-1]$
 
$\int \limits_0^1 xe^xdx=\frac{1}{2}[e-1]$

 

answered Feb 11, 2013 by meena.p
 
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