Integral: ∫(exp(6x))/(exp(12x)+25)dx Solution

[suprabay]

Homework Statement

∫(exp(6x))/(exp(12x)+25)dx

Homework Equations

answer: -arctan[5/exp(6*x)]/30

The Attempt at a Solution

honestly, don't know where to start. i was looking at another forum and tried to set u=exp(x) du=exp(x) and dx=du/u. plugging that in i got u^6/(u^12+25)*du/u. not sure where to go from there or if that is even the way to go.

 

[gabbagabbahey]

Hi supraboy, Welcome to PF!

Try the substitution $$u=e^{6x}$$ instead

 

[VKint]

Try setting u = e^6x. Then du = 6e^6x and e^12x = u².

 

[suprabay]

ok, setting u=e^6x du=6e^6x, then dx=du/6u?

then, it would be int(u/u^2+25)du

using the formula int(a^2+u^2) = (1/a)arctan(u/a) + C

i get, (1/5)arctan(e^6x/5)dx or (1/30)arctan(e^6x/5) + C

this is incorrect though because the answer is negative and it should be arctan(5/e^6x) instead of arctan(e^6x/5).

any ideas?

 

[VKint]

You made a mistake with the substitution. Write your integral like this:
$$\int \frac{(e^{6x} dx)}{(e^{6x})^2 + 25}$$
If $$u = e^{6x}$$, then $$e^{6x} dx = \frac{1}{6} du$$. Try working from there.