Mathematics

$$\displaystyle \int (e^\sqrt[3]{x}dx)=$$


ANSWER

$$3(x^{2/3} - 2x^{1/3}+2)\exp(\sqrt[3]{x})+c$$


SOLUTION
$$I= \int { { e }^{ \sqrt [ 3 ]{ x }  }dx } $$
Put $$\sqrt [ 3 ]{ x } =t$$
$$\displaystyle  \dfrac { 1 }{ 3 } { x }^{ -2/3 }dx=dt$$
$$\displaystyle dx=\dfrac { 3 }{ { t }^{ -2 } } dt$$

So, $$I=3\int { { { t }^{ 2 }e }^{ t }dt } $$
$$=3[{ { t }^{ 2 }e }^{ t }-2\int { t{ e }^{ t }dt } ]+c$$
$$=3{ { t }^{ 2 }e }^{ t }-6[t{ e }^{ t }-\int { { e }^{ t }dt } ]+c$$
$$ =3{ { t }^{ 2 }e }^{ t }-6t{ e }^{ t }+6{ e }^{ t }+c$$
$$=3(x^{2/3} - 2x^{1/3}+2)e^{\sqrt[3]{x}}+c$$
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Single Correct Medium Published on 17th 09, 2020
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