Question

Integral of $x^2{e}^{x}dx$ will be?

Answer 1

Find the integral

Given, $x^2{e}^x$

Now,

We know that , $\int u.vdx=u\int vdx-\int \left\{\frac{du}{dx}\int vdx\right\}dx$

Applying integration by parts,

Here ,

$$\begin{gathered} u=x^2,v=e^x \\ {u}^{\text{'}}=2x,\int vdx=e^x \end{gathered}$$

According to the formula $\int u.vdx=u\int vdx-\int \left\{\frac{du}{dx}\int vdx\right\}dx$ we get,

$$\begin{gathered} u=x^2,v=e^x \\ {u}^{\text{'}}=2x,\int vdx=e^x \\ \Rightarrow \int x^2{e}^{x}dx=x^2{e}^x-\int 2x{e}^{x}dx..............\left(1\right) \end{gathered}$$

Again applying integrating by parts on the function $2x{e}^x$,

Here ,

$$\begin{gathered} u=x,v=e^x \\ {u}^{\text{'}}=1,\int vdx=e^x \\ \Rightarrow \int 2x{e}^{x}dx=2x{e}^x-2e^x+c............\left(2\right) \end{gathered}$$

Now , By using the equation $\left(1\right)$ and equation $\left(2\right)$ we get,

$\int x^2{e}^{x}dx=x^2{e}^x-2x{e}^x+2e^x+c$

Hence , integral of $x^2{e}^{x}dx$ is $x^2{e}^x-2x{e}^x+2e^x+c$..