Mathematics

$$\int {x^2}\,{e^x}\, dx =?$$


SOLUTION
$$I=\displaystyle\int{{x}^{2}{e}^{x}dx}$$

Integrating by parts, we get

Let $$u={x}^{2}\Rightarrow\,du=2x\,dx$$

$$dv={e}^{x}dx\Rightarrow\,v={e}^{x}$$

$$I={x}^{2}{e}^{x}-\displaystyle\int{2x{e}^{x}dx}$$

$$I={x}^{2}{e}^{x}-2\displaystyle\int{x{e}^{x}dx}$$

Let $$u=x\Rightarrow\,du=dx$$

$$dv={e}^{x}dx\Rightarrow\,v={e}^{x}$$

$$I={x}^{2}{e}^{x}-2\left[x{e}^{x}-\displaystyle\int{{e}^{x}dx}\right]$$

$$I={x}^{2}{e}^{x}-2\left[x{e}^{x}-{e}^{x}\right]+c$$

$$\therefore\,I={x}^{2}{e}^{x}-2x{e}^{x}+2{e}^{x}+c$$
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Subjective Medium Published on 17th 09, 2020
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