Question

Evaluate the definite integrals: $\int^1_0 {x\ e^{x^2}}\, dx$

Answer 1

Solution:

$\int^1_0 {x\ e^{x^2}}\, dx$

let

${x}^{2} = t \\ \\ 2x \: dx = dt \\ \\ xdx = \frac{dt}{2}$
Substitute the value

$\int^1_0 {\ e^{t}} \frac{dt}{2} \\ \\ = \frac{1}{2} \int^1_0 {\ e^{t}}dt \\ \\ = \frac{1}{2} \bigg[ {e}^{t} \bigg]^\bm1_{0} \\ \\ = \frac{1}{2} {e}^{1} - \frac{1}{2} \\ \\ \int^1_0 {x\ e^{x^2}}\, dx \: = \frac{1}{2} ( {e}^{1} - 1)$
Hope it helps you.