How do you evaluate the integral of #int 3x^5 e^x dx#?

Question

Isaiah Allen · Accepted Answer

#3(x^5 e^x - 5( x^4 e^x - 4( x^3 e^x-3( x^2e^x -2 (x-1)e^x))))+C#

or

#3 ( x^5 - 5 x^4 + 20 x^3 - 60 x^2 + 120 x - 120)e^x + C# In this development we will consider the integration constant #C = 0#. Calling #I_n = int x^n e^x dx # and after #d/(dx)(x^n e^x) = n x^{n-1}e^x + x^n e^x# we have #int d/(dx)(x^n e^x)dx = n int x^{n-1}e^x dx + int x^n e^x dx# or #x^n e^x= n I_{n-1}+I_n# with #I_0 = int x^0 e^x dx = int e^x dx = e^x# so we can iterate #I_0+I_1 = x e^x-> I_1 = x e^x - e^x = (x-1)e^x# #2I_1+I_2 = x^2 e^x->I_2 = x^2e^x -2 (x-1)e^x # #3I_2+I_3 = x^3 e^x->I_3 = x^3 e^x-3( x^2e^x -2 (x-1)e^x)# #4I_3+I_4 = x^4 e^x->I_4 = x^4 e^x - 4( x^3 e^x-3( x^2e^x -2 (x-1)e^x))# and finally #5I_4+I_5 = x^5 e^x# obtaining #I_5 = x^5 e^x - 5( x^4 e^x - 4( x^3 e^x-3( x^2e^x -2 (x-1)e^x)))# Our integral is #3I_5# so #int 3x^5 e^x dx = 3I_5#

Ezekiel Alexander · Answer

undefined To evaluate the integral ∫ 3x^5 e^x dx, you can use integration by parts method. Let u = 3x^5 and dv = e^x dx. Then, differentiate u to find du and integrate dv to find v. Finally, apply the integration by parts formula:

∫ u dv = uv - ∫ v du

Substitute the values of u, v, du, and dv into the formula and solve the integral.