Question

How do you evaluate the integral of `[ (e^7x) / ((e^14x)+9) dx ]`?

Answer 1

`int{{ax}/{bx+c}}dx={ac}/b^2(b/cx-ln|b/cx+1|)+C`

where `a`, `b`, and `c` are constants. `C` is the constant of integration.
I got the above result by the following process:

`int {{ax}/{bx+c}}dx=int{a/{b}{b/cx}/{b/cx+1}}dx`

Let `u=b/cx`, Then `dx=c/bdu`.

By substituting, the integral in terms of `u` is,

`{ac}/b^2int{u/{u+1)}du={ac}/b^2int{1-1/{u+1}}du`

`={ac}/b^2(u-ln|u+1|)+C`

Now that the integration is done, express the result in terms of `x`. Substitute `u=b/cx` and get

`={ac}/b^2(b/cx-ln|b/cx+1|)+C`

Answer 2

I have another method

Substitute `u = e^(7x)`

`du = 7e^(7x)dx`

`int(e^(7x))/(e^(14x)+9)dx = 1/7int(7e^(7x))/(e^(14x)+9)dx`

`u^2=e^(14x)`

`1/7int1/(u^2+9)du = 1/63int1/(1/9u^2+1)du`

Substitute again, `t = 1/3u` `t^2=1/9u^2`

`1/21int1/(t^2+1)dt`

`1/21[arctan(t)]+C`

Substitute back

`1/21arctan(1/3e^7x)+C`