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

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Answer 1 · 2015-05-04T04:20:03

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

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 · 2015-05-13T15:46:12

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

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

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