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Evaluate the definite integral\[\int\limits_0^1x\;e^{x^2}dx\]

1 Answer

  • (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
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,
But $e^0=1$
$\int \limits_0^1 xe^xdx=\frac{1}{2}[e-1]$


answered Feb 11, 2013 by meena.p