`int_0^oo e^x/(1+e^x) dx` Determine whether the integral diverges or converges. Evaluate the integral if it converges.

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`int_0^infty e^x/(1+e^x)dx=`

Substitute `u=1+e^x` `=>` `du=e^xdx,` `u_l=1+e^0=2,` `u_u=lim_(x to infty)(1+e^x)=infty.` 

 `int_1^infty 1/u du=ln |u||_2^infty=lim_(u to infty)ln u-ln 2=infty`

As we can see the integral diverges.                                                  ...

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`int_0^infty e^x/(1+e^x)dx=`

Substitute `u=1+e^x` `=>` `du=e^xdx,` `u_l=1+e^0=2,` `u_u=lim_(x to infty)(1+e^x)=infty.` 

 `int_1^infty 1/u du=ln |u||_2^infty=lim_(u to infty)ln u-ln 2=infty`

As we can see the integral diverges.                                                        Image below shows graph of the function and area under it (value of the integral). As we can see from the image the function converges to 1, but the area under the graph increases indefinitely (integral diverges). 

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