Question

Evaluate the given integral using the substitution (or method) indicated.

(x + 7)e^(x + 7)^(2) dx; u = (x + 7)^(2)

Answer 1

int(X^2*e^(-x))dx
Substitute
s=-x
ds=-1dx
= - int(e^s*u^2) ds
Integrate by parts
u=s^2
dv=e^s ds
du=2s ds
v=e^s

= int 2e^s * s ds - e^s * s^2

Factor out the 2:

2 int [ e^s * s ds - e^s s^2

Integrate by parts
u=s
dv=e^s ds
du=1 ds
v=e^s

becomes = -e^s * s^2 * s - 2 int s^2 ds

The integral of e^s is e^s
becomes:
= - e^s * s^2 + 2 e^s * s - 2 e^s + C

Substitute back in s=-x
= - e^(-s) * x^2 - 2 e^(-s) * x - 2 e^(-x) + C

Answer 2

S isn't a part of it, it's just the symbol i couldn't copy and paste