Mathematics

Integrate:
$$2x^2e^{x^2}$$


SOLUTION
$$2\int { { x }^{ 2 }{ e }^{ { x }^{ 2 } }dx } =2\left[ { x }^{ 2 }\int { { e }^{ { x }^{ 2 } }dx } +\int { \dfrac { d }{ dx } { x }^{ 2 }. } \int { { e }^{ { x }^{ 2 } }dx }  \right] dx\quad \text(integration\quad by\quad parts)\\ =\dfrac { 2{ x }^{ 2 }{ e }^{ { x }^{ 2 } } }{ 2x } +2\int { 2x.\dfrac { { e }^{ { x }^{ 2 } } }{ 2x } dx } \\ =x{ e }^{ { x }^{ 2 } }+\dfrac { 2.{ e }^{ { x }^{ 2 } } }{ 2x } +C\\ =\left( \dfrac { { x }^{ 2 }+1 }{ x }  \right) { e }^{ { x }^{ 2 } }+C$$
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Subjective Medium Published on 17th 09, 2020
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