Evaluate the definite integral _ 0 ^ 1 x e ^ x ^ 2 d x. - Vidyadip

Question

Evaluate the definite integral $\int _ { 0 } ^ { 1 } x e ^ { x ^ { 2 } } d x.$

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Answer

Let $I = \int _ { 0 } ^ { 1 } x e ^ { x ^ { 2 } } d x$
Put $x ^ { 2 } = t \Rightarrow 2 x d x = d t \Rightarrow d x = \frac { d t } { 2 x }$
Lower limit when x = 0, then t = 0
Upper limit when x = 1, then t = 1.
$\therefore \quad I = \int _ { 0 } ^ { 1 } x e ^ { t } \frac { d t } { 2 x } = \frac { 1 } { 2 } \int _ { 0 } ^ { 1 } e ^ { t } d t$
$= \frac { 1 } { 2 } \left[ e ^ { t } \right] _ { 0 } ^ { 1 } = \frac { 1 } { 2 } \left[ e ^ { 1 } - e ^ { 0 } \right] = \frac { 1 } { 2 } [ e - 1 ]$

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