Question #61a6c

1 Answer
Apr 12, 2017

Always let "u" equal the polynomial because differentiating it makes the resulting integral simpler

Explanation:

Integration by parts:

#intudv = uv-intvdu#

let #u = x^2+2x+1#, then #du = 2x+2dx#
let #dv=e^(7x)dx#, then #v = 1/7e^(7x)#

#int(x^2+2x+1)e^(7x)dx= 1/7(x^2+2x+1)e^(7x)-1/7int(2x+2)e^(7x)dx#

Repeat the integration by parts:

#int(x^2+2x+1)e^(7x)dx= 1/7(x^2+2x+1)e^(7x)-1/7int(2x+2)e^(7x)dx#

let #u = 1/7(2x+2)#, then #du=2dx#
let #dv=e^(7x)dx#, then #v = 1/7e^(7x)#

#int(x^2+2x+1)e^(7x)dx= 1/7(x^2+2x+1)e^(7x)-1/49(2x+2)e^(7x)+2/49inte^(7x)dx#

#int(x^2+2x+1)e^(7x)dx= 1/7(x^2+2x+1)e^(7x)-1/49(2x+2)e^(7x)+2/343e^(7x)+C#