Question

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

Answer 1

$2x^3{e}^{x^2}dx$

$\Rightarrow$ Let $e^{x^2}=t$

$\Rightarrow$ $2x{e}^{x^2}dx=dt$

$\Rightarrow$ So, $\int 2x^3{e}^{x^2}dx=\int x^2.2x.e^{x^2}dx$

$=\int {\log }_{e}tdt$ [ Since $e^{x^2}=t\Rightarrow x^2={\log }_{e}t$ ]

$={\log }_{e}t.\int 1.dt-\int (\frac{d\left({\log }_{e}t\right)}{dt}.\int 1.dt)dt$

$={\log }_{e}t.t-\int \frac{1}{t}.tdt$

$=t{\log }_{e}t-\int 1.dt$

$=t{\log }_e-t+c$

$=t({\log }_{e}t-1)+c$

$=e^{x^2}({\log }_e{e}^{x^2}-1)+c$

$=e^{x^2}(x^2-1)+c$