Evaluate each of the following integral: `int_0^1x\ e^x^2dx`

Explanation

`I=int_{0}^{1} x e^{x^{2}} d x`

`=frac{1}{2} int_{0}^{1} e^{x^{2}} 2 x d x`

Put `x^{2}=z`

`Rightarrow 2 x d x=d z`

When `x rightarrow 0, z rightarrow 0`

When `x rightarrow 1, z rightarrow 1`

`therefore I=frac{1}{2} int_{0}^{1} e^{z} d z`

`=frac{1}{2} times. e^{z}|_{0} ^{1}`

...