Reduction Formula

∫u dv = u v - ∫v du

For any positive integer n,
∫x^n e^x dx　　(Note: All the "from 0 to 1" are omitted in the integral signs below.)
= ∫x^n d(e^x)
= x^n (e^x) |{from 0 to 1} - ∫e^x d(x^n)
= e - n ∫x^(n-1) e^x dx

Therefore,
∫x^5 e^x dx
= e - 5 ∫x^4 e^x dx
= e - 5e + 5•4∫x^3 e^x dx
= e - 5e + 5•4 e - 5•4•3∫x^2 e^x dx
= e - 5e + 5•4 e - 5•4•3 e + 5•4•3•2 ∫x e^x dx
= e - 5e + 5•4 e - 5•4•3 e + 5•4•3•2 e - 5•4•3•2•1 ∫e^x dx
= e - 5e + 5•4 e - 5•4•3 e + 5•4•3•2 e - 5•4•3•2•1 [e^x] |{from 0 to 1}
= e - 5e + 5•4 e - 5•4•3 e + 5•4•3•2 e - 5•4•3•2•1 (e - 1)
= e - 5e + 20e - 60e + 120e - 120e + 120
= -44e + 120.