Question

Evaluate each of the following integrals:

${\int }_0^{1}x{e}^{x^2}dx$ [CBSE 2014]

Answer 1

$$\begin{gathered} I={\int }_0^{1}x{e}^{x^2}dx \\ =\frac{1}{2}{\int }_0^1{e}^{x^2}2xdx \end{gathered}$$

Put $x^2=z$

$\Rightarrow 2xdx=dz$

When $x\to 0,z\to 0$

When $x\to 1,z\to 1$

$$\begin{gathered} \therefore I=\frac{1}{2}{\int }_0^1{e}^{z}dz \\ =\frac{1}{2}\times {\overline{)e^z}}_0^1 \\ =\frac{1}{2}\left(e-e^0\right) \\ =\frac{1}{2}\left(e-1\right) \end{gathered}$$