Question

\[\int2 x^3 e^{x^2} dx\]

Answer 1

\[\int2 x^3 \cdot e^{x^2} dx\]
\[ = \int x^2 \cdot \left( e^{x^2} \right) \cdot \text{ 2x dx }\]
`  \text{  Let }  x^2" = t `

\[ \Rightarrow \text{ 2x dx } = dt\]
\[ = \int t_I \cdot {e_{II}}^t dt\]
\[ = t \cdot e^t - \int1 \cdot e^t dt\]
\[ =  \text{ t   e}^t - e^t + C\]
\[ = \text{ x}^2   \text{ e}^{x^2} - e^{x^2} + C\]
\[ = e^{x^2} \left( x^2 - 1 \right) + C\]