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