The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,dx$$     is-

Solution :
$$\eqalign{ & \int_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,dx \cr & = \int_0^1 {\left( {1 + 1 - \frac{{{x^2}}}{{1!}} + \frac{{{x^4}}}{{2!}} - \frac{{{x^6}}}{{3!}} + .....\infty } \right)} dx \cr & = \left[ {2x - \frac{{{x^3}}}{{3.1!}} + \frac{{{x^5}}}{{5.2!}} - \frac{{{x^7}}}{{7.3!}} + .....\infty } \right]_0^1 \cr & = \left[ {2 - \frac{1}{{3.1!}} + \frac{1}{{5.2!}} - \frac{1}{{7.3!}} + .....\infty } \right] \cr} $$