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Integrate the function\[x^2e^x\]

1 Answer

  • (i)Let us consider two functions u and v and if they are of the form $\int u dv,$then we can solve it by the method of integration by parts\[\int udv=uv-\int vdu\]
  • (ii)$\int e^xdx=e^x+c.$
Given $I=\int x^2e^x dx.$
Clearly the given integral function is of the form $\int u dv$,so let us follow the method of integration by parts where \[\int udv=uv-\int vdu\]
Let $u=x^2$.
On differentiating we get
Let $dv=e^xdx.$
On integrating on both sides we get
On substituting for u,v,du and dx we get,
$\int x^2e^xdx=(x^2.e^x)-\int e^x.2xdx.$
$\;\;\;=(x^3e^x)-2\int e^x.xdx.$-----(1)
Again $\int e^xdx $ is of the form $\int u dv.$
Let u=x.
On differentiating we get
let $dv=e^xdx.$
On integrating we get
Hence on substituting for u,v,du and dv we get
$\int x.e^xdx=(xe^x)-\int e^xdx.$
On integrating we get
Substituting this in equ(1) we get
$\int x^2e^xdx=x^2e^x-2(xe^x-e^x)+c.$
Taking $e^x$ as the common factor we get


answered Feb 8, 2013 by sreemathi.v