Integrals: int e^x/ e^2x + 9dx, int e^2x + 9/ e^xdx

[Guest]

I need help with the following integrals:

the integral of e^x/ e^2x + 9dx

and the integral of e^2x + 9/ e^xdx

 

[galactus]

Re: Integrals

I need help with the following integrals:

the integral of e^x/ e^2x + 9dx

Please use grouping symbols. Your post is ambiguous.

I assume you mean: \(\displaystyle \L\\\int\frac{e^{x}}{e^{2x}+9}dx\)

If so, let \(\displaystyle u=e^{x}, \;\ du=e^{x}dx\)

That gives you:

\(\displaystyle \L\\\int\frac{1}{u^{2}+9}du\)

Does this look like a familiar integral?. Think arctan

and the integral of e^2x + 9/ e^xdx

Is that \(\displaystyle \L\\\int\frac{e^{2x}+9}{e^{x}}dx\)

If so, rewrite as \(\displaystyle \L\\e^{x}+9e^{-x}\) and integrate.

 

[arthur ohlsten]

e^x dx / [e^2x+9]

looks similiar to a trig integral
rewrite
e^x dx / [9 [ e^2x/9+1] ]

let z= e^x / 9
dz= 1/9 e^x dx
rewrite

9 [1/9 e^x dx] / 9[e^2x/9 +1]
substitute
dz / [[z/3]^2+1]
you can now integrate

-1/3 tan^-1 [3e^x) +c answer
please check for errors

====================================================
[e^2x+9] dx /e^x rewrite
e^2x dx/e^x +9dx/e^x rewrite
e^x dx +9 e^-x dx
you can now integrate

e^x -9e^-x +c answer
please check for errors

Arthur