Mathematics

Solve $$\int {2{x^3}{e^{{x^2}}}} $$


ANSWER

$$(x^2e^{x^2}-e^{x^2})+C$$


SOLUTION
$$\int { 2{ x }^{ 3 }{ e }^{ { x }^{ 2 } } } dx$$
by integration by partly, we have
Let $${ x }^{ 2 }=t$$
$$\Rightarrow 2xdx=dt$$
$$\Rightarrow 2\sqrt { t } dx=dt$$
$$\Rightarrow \int { 2{ x }^{ 2 }.x.{ e }^{ { x }^{ 2 } }dx } $$
$$\Rightarrow \int { 2t } { e }^{ t }.\dfrac { dt }{ 2 } $$
$$\Rightarrow \int { t } .{ e }^{ t }dt=t.\int { { e }^{ t }dt } -\int { \dfrac { d }{ dt } \left( t \right)  } \int { { e }^{ t }dt } $$
                     $$=t.{ e }^{ t }-\int { { e }^{ t } } dt$$
                     $$=t.{ e }^{ t }-{ e }^{ t }+c$$
                     $$=\left( { x }^{ 2 }.{ e }^{ { x }^{ 2 } }-{ e }^{ { x }^{ 2 } } \right) +c$$
Hence, the answer is $$\left( { x }^{ 2 }.{ e }^{ { x }^{ 2 } }-{ e }^{ { x }^{ 2 } } \right) +c.$$
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Single Correct Medium Published on 17th 09, 2020
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