A Very Nice Integral Problem

A Very Nice Integral Problem
calculus, integral, integration

[@ParthMoghe09]

Hello, we can use the DI method here to save a ton of time, it is basically integration by parts, but better. i got the answer within 15 seconds using that.

[@gelbkehlchen]

Solution: ∫(x²-2x+5)*e^(-x)*dx = ------------------------------------ Solution by partial integration: Partial integration can be derived from the product rule of differential calculus. The product rule of differential calculus states: (u*v)’ = u’*v+u*v’ |-u’*v ⟹ (u*v)’-u’*v = u*v’ ⟹ u*v’ = (u*v)’-u’*v |∫() ⟹ ∫u*v’*dx = u*v-∫u’*v*dx ------------------------------------ = -(x²-2x+5)*e^(-x)+2*∫(x-1)*e^(-x)*dx = -(x²-2x+5)*e^(-x)+2*[-(x-1)*e^(-x)+∫e^(-x)*dx] = = -(x²-2x+5)*e^(-x)+2*[-(x-1)*e^(-x)-e^(-x)]+C = = (-x²+2x-5)*e^(-x)+2*e^(-x)*[-x+1-1]+C = = (-x²+2x-5)*e^(-x)-2*e^(-x)*x+C = = (-x²+2x-5-2x)*e^(-x)+C = = (-x²-5)*e^(-x)+C = = -(x²+5)*e^(-x)+C Checking the result by deriving: [-(x²+5)*e^(-x)+C]’ = -2x*e^(-x)+(x²+5)*e^(-x) = (x²-2x+5)*e^(-x) Everything okay!

[@wasimvillidad3000]

Let u = -x and use the fact that ∫(f(x) + f'(x))e ͯdx = f(x)e ͯ + c