Question

The Integration of
${x}^{2} {e}^{3x} dx$
is


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Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:\displaystyle\int\rm {x}^{2} {e}^{3x} \: dx$

We know,

Integration by parts

$\boxed{\tt{ \displaystyle\int\rm uv \: dx \: = u\displaystyle\int\rm vdx - \displaystyle\int\rm \bigg[\dfrac{d}{dx}u\displaystyle\int\rm vdx \bigg]dx \: }}$

So, here

$\purple{\rm :\longmapsto\:u = {x}^{2}}$

$\purple{\rm :\longmapsto\:v = {e}^{3x} }$

So, using the above result, we have

$\rm \:  =  \: {x}^{2} \displaystyle\int\rm {e}^{3x}dx - \displaystyle\int\rm \bigg[\dfrac{d}{dx} {x}^{2} \displaystyle\int\rm {e}^{3x}dx \bigg]dx$

$\rm \:  =  \: {x}^{2}\dfrac{{e}^{3x}}{3} - \displaystyle\int\rm 2x\dfrac{{e}^{3x}}{3}dx$

$\rm \:  =  \: \dfrac{ {x}^{2}{e}^{3x}}{3} - \dfrac{2}{3}\displaystyle\int\rm x{e}^{3x} \: dx$

Again, using integration by parts,

$\rm \:  =  \: \dfrac{ {x}^{2}{e}^{3x}}{3} - \dfrac{2}{3}\bigg(x\displaystyle\int\rm {e}^{3x} \: dx - \displaystyle\int\rm \bigg[\dfrac{d}{dx}x\displaystyle\int\rm {e}^{3x}dx \bigg]dx\bigg)$

$\rm \:  =  \: \dfrac{ {x}^{2}{e}^{3x}}{3} - \dfrac{2}{3}\bigg(\dfrac{x{e}^{3x}}{3} - \displaystyle\int\rm \frac{{e}^{3x}}{3} dx\bigg)$

$\rm \:  =  \: \dfrac{ {x}^{2}{e}^{3x}}{3} - \dfrac{2}{3}\bigg(\dfrac{x{e}^{3x}}{3} - \dfrac{{e}^{3x}}{9} \bigg) + c$

$\rm \:  =  \: \dfrac{ {x}^{2}{e}^{3x}}{3} - \dfrac{2x{e}^{3x}}{9} + \dfrac{2{e}^{3x}}{27} + c$

Hence,

$\boxed{\tt{ \rm \: \displaystyle\int\rm {x}^{2}{e}^{3x}dx =  \: \dfrac{ {x}^{2}{e}^{3x}}{3} - \dfrac{2x{e}^{3x}}{9} + \dfrac{2{e}^{3x}}{27} + c}}$

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Formula Used

$\purple{\rm :\longmapsto\:\boxed{\tt{ \: \: \: \frac{d}{dx} {x}^{n} = {nx}^{n - 1} \: \: \: }}}$

$\purple{\rm :\longmapsto\:\boxed{\tt{ \: \: \: \displaystyle\int\rm {e}^{ax + b}dx = \frac{{e}^{ax + b}}{b} + c \: \: \: }}}$

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$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$