Question

I would absolutely love and adore some help integrating (6e^6t)/(e^(12t) +12 e^(6t) +27) dx

Answer 1

\[
I=\int\frac{6e^{6t}}{e^{12t}+12e^{6t}+27}dt
\]
start with the substitution
\[e^{6t}=u\]

Answer 2

so I should get to:
\[I=\int\limits_\ \frac{u}{u^{2}+12u+27} du\] correct?

Answer 3

which doesn't look scary at all

Answer 4

yes. factorize the denominator

Answer 5

Here's where I'm at so far:\[=6 \int\limits_{} \frac{u}{(u+9)(u+3)} du\]
\[\therefore\frac{u}{(u+9)(u+3)}=\frac{A}{u+9}+\frac{B}{u+3}\]
\[\therefore u=A(u+3)+B(u+9)\]

Answer 6

set u=-9
\(-9=A(-9+3)\implies A={3\over2}\)
set a=-3
\(-3=C(-3+9)\implies B=-{1\over2}\)