use u subsitution
the derivitive of eˣ+2 is eˣ
u=eˣ+2
du/dx=eˣ
du=eˣdx
we see a eˣdx in the problem
[tex] \int\limits (e^x+2)^3e^x} \, dx [/tex]
sub du for that and u for eˣ+2
[tex] \int\limits {u^3} \, du [/tex]
remember
[tex] \int\limits {u^x} \, du= \frac{u^{x+1}}{x+1} [/tex]
[tex] \int\limits {u^3} \, du= \frac{u^{3+1}}{3+1}= \frac{u^4}{4} [/tex]
sub eˣ+2 for u
[tex] \frac{(e^x+2)^4}{4} [/tex]
don't forget to add a constant since the derivitive of a constant is 0
the original function is [tex] \frac{(e^x+2)^4}{4} +C[/tex] where C is a constant
