SOLUTION: What is the integral of dx/(e^(2x)+1) using trigonometric substitution?

Notice that  can be written as the sum of squares:

 

Draw a right triangle
Always put the simpler expression on the adjacent side.  Since 1
is simpler than ex, we put it on the adjacent side.  
The sign between the squares is + so we put the more complicated 
expression on the opposite side.  [If it had been - we would have 
put the more complicated expression on the hypotenuse.]



So 


Solve for x by taking natural logs of both sides:

We find dx by multiplying the derivative by dθ 



Then we use the Pythagorean theorem to label the
missing side, which is the hypotenuse.

 

Now we're ready to substitute:



We substitute for dx



Now here's a trick.  Observe that the denominator 
equals the square of the hypotenuse, and since 

the denominator is just sec2(θ)



Now the sec2(θ)'s cancel:









Now we look back at the triangle and find that











   <-- final answer

Edwin