Question

Question #61a6c

Answer 1

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`