How do you integrate #int x^3 e^x dx # using integration by parts?

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Answer 1 · 2016-03-01T18:38:18

#\int x^3e^xdx=x^3e^x-3(x^2e^x-2(e^x x-e^x))+C#

#\int x^3e^xdx#

Applying integration by parts as: #\int uv'=uv-\int u'v#
#u=x^3,u'=3x^2,v'=e^x,v=e^x#

#=x^3e^x-\int 3x^2e^xdx#

#\int 3x^2e^xdx=3(x^2e^x-2(e^x x-e^x))# as under

=#\int 3x^2e^xdx#

taking the constant out as: #\int a\cdot f(x)dx=a\cdot \int f(x)dx#

#=3\int x^2e^xdx#

Applying integration by parts as: #\int uv'=uv-\int u'v#

#u=x^2,u'=2x,v'=e^x,v=e^x#

#=3(x^2e^x-\int2xe^xdx)#

taking the constant out as: #\int a\cdot f(x)dx=a\cdot \int f(x)dx#

#=3(x^2e^x-2\int xe^xdx)#

Applying integration by parts as: #\int uv'=uv-\int u'v#

#u=x,u'=1,v'=e^x,v=e^x#

#=3(x^2e^x-2(xe^x-\int 1e^xdx))#

#=3(x^2e^x-2(e^x x-\int e^xdx))#

Using the common integral #\inte^xdx=e^x#
#=3(x^2e^x-2(e^x x-e^x))#

#=x^3e^x-3(x^2e^x-2(e^x x-e^x))#

Adding a constant to the solution,

#=x^3e^x-3(x^2e^x-2(e^x x-e^x))# +C