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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