Mathematics

# Evaluate: $\displaystyle \int_0^x x^6 e^{\frac{-x}{2}} dx =$

##### SOLUTION
Consider $\int{{x}^{6}{e}^{\frac{-x}{2}}dx}$
Let $t=\dfrac{-x}{2}\rightarrow dt=\dfrac{-1}{2}dx$
We have $t=\dfrac{-x}{2}\Rightarrow 2t=-x$ or ${\left(2t\right)}^{6}={\left(-x\right)}^{6}$ or $64{t}^{6}={x}^{6}$
Now,$\int{{x}^{6}{e}^{\frac{-x}{2}}dx}=-64\int{{t}^{6}{e}^{t}2dt}$
$=-128\int{{t}^{6}{e}^{t}dt}$            ........$\left(1\right)$
Consider $\int{{t}^{6}{e}^{t}dt}$
Let $u={t}^{6}\Rightarrow du=6{t}^{5}dt$ and $dv={e}^{t}dt\Rightarrow v={e}^{t}$
$\int{{t}^{6}{e}^{t}dt}={t}^{6}{e}^{t}-6\int{{t}^{5}{e}^{t}dt}$
Consider $\int{{t}^{5}{e}^{t}dt}$
Let $u={t}^{5}\Rightarrow du=5{t}^{4}dt$ and $dv={e}^{t}dt\Rightarrow v={e}^{t}$
$\int{{t}^{5}{e}^{t}dt}={t}^{5}{e}^{t}-5\int{{t}^{4}{e}^{t}dt}$       ........$\left(2\right)$
Substituting $\left(2\right)$ in $\left(1\right)$ we get
$-128\int{{t}^{6}{e}^{t}dt}=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\int{{t}^{4}{e}^{t}dt}\right)\right]$           .........$\left(3\right)$
Consider $\int{{t}^{4}{e}^{t}dt}$
Let $u={t}^{4}\Rightarrow du=4{t}^{3}dt$ and $dv={e}^{t}dt\Rightarrow v={e}^{t}$
$\int{{t}^{4}{e}^{t}dt}={t}^{4}{e}^{t}-4\int{{t}^{3}{e}^{t}dt}$       .........$\left(4\right)$
Substituting $\left(4\right)$ in $\left(3\right)$ we get
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\left({t}^{4}{e}^{t}-4\int{{t}^{3}{e}^{t}dt}\right)\right)\right]$         ......$\left(5\right)$
Consider $\int{{t}^{3}{e}^{t}dt}$
Let $u={t}^{3}\Rightarrow du=3{t}^{2}dt$ and $dv={e}^{t}dt\Rightarrow v={e}^{t}$
$\int{{t}^{3}{e}^{t}dt={t}^{3}{e}^{t}-3\int{{t}^{2}{e}^{t}}dt}$    .........$\left(6\right)$
Substituting   $\left(6\right)$ in $\left(5\right)$ we get
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\left({t}^{4}{e}^{t}-4\left({t}^{3}{e}^{t}-3\int{{t}^{2}{e}^{t}dt}\right)\right)\right)\right]$      .........$\left(7\right)$
Consider $\int{{t}^{2}{e}^{t}dt}$
Let $u={t}^{2}\Rightarrow du=2tdt$ and $dv={e}^{t}dt\Rightarrow v={e}^{t}$
$\int{{t}^{2}{e}^{t}dt}={t}^{2}{e}^{t}-2\int{t{e}^{t}dt}$         .........$\left(8\right)$
Substituting $\left(8\right)$ in $\left(7\right)$ we get
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\left({t}^{4}{e}^{t}-4\left({t}^{3}{e}^{t}-3\left({t}^{2}{e}^{t}-2\int{t{e}^{t}dt}\right)\right)\right)\right)\right]$      .........$\left(9\right)$
Consider  $\int{t{e}^{t}dt}$
Let $u=t\Rightarrow du=dt$ and $dv={e}^{t}dt\Rightarrow v={e}^{t}$
$\int{t{e}^{t}dt}=t{e}^{t}-\int{{e}^{t}dt}=t{e}^{t}-{e}^{t}$          .........$\left(10\right)$
Substituting $\left(10\right)$ in $\left(9\right)$ we get
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\left({t}^{4}{e}^{t}-4\left({t}^{3}{e}^{t}-3\left({t}^{2}{e}^{t}-2\left(t{e}^{t}-{e}^{t}\right)\right)\right)\right)\right)\right]$
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\left({t}^{4}{e}^{t}-4\left({t}^{3}{e}^{t}-3\left({t}^{2}{e}^{t}-2\left(t{e}^{t}-{e}^{t}\right)\right)\right)\right)\right)\right]+c$
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\left({t}^{4}{e}^{t}-4\left({t}^{3}{e}^{t}-3\left({t}^{2}{e}^{t}-2t{e}^{t}+2{e}^{t}\right)\right)\right)\right)\right]+c$
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\left({t}^{4}{e}^{t}-4\left({t}^{3}{e}^{t}-3{t}^{2}{e}^{t}+6t{e}^{t}-6{e}^{t}\right)\right)\right)\right]+c$
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5\left({t}^{4}{e}^{t}-4{t}^{3}{e}^{t}+12{t}^{2}{e}^{t}-24t{e}^{t}+24{e}^{t}\right)\right)\right]+c$
$=-128\left[{t}^{6}{e}^{t}-6\left({t}^{5}{e}^{t}-5{t}^{4}{e}^{t}+20{t}^{3}{e}^{t}-60{t}^{2}{e}^{t}+120t{e}^{t}-120{e}^{t}\right)\right]+c$
$=-128\left[{t}^{6}{e}^{t}-6{t}^{5}{e}^{t}+30{t}^{4}{e}^{t}-120{t}^{3}{e}^{t}+360{t}^{2}{e}^{t}-720t{e}^{t}+720{e}^{t}\right]_{0}^{x}+c$
$=-128\left[{x}^{6}{e}^{x}-6{x}^{5}{e}^{x}+30{x}^{4}{e}^{x}-120{x}^{3}{e}^{x}+360{x}^{2}{e}^{x}-720x{e}^{x}+720{e}^{x}-720\right]+c$
where $c$ is the constant of integration.

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Subjective Medium Published on 17th 09, 2020
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