`int e^x/((e^(2x)+1)(e^x-1)) dx` Use substitution and partial fractions to find the indefinite integral

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Apply integral substitution:`u=e^x`



Now let's create partial fraction template for the integrand,


Multiply the equation by the denominator,




Equating the coefficients of the like terms,

`A+B=0`    -------------------------(1)

`-B+C=0`  -----------------------(2)

`A-C=1`       -----------------------(3)

Now we have to solve the above three linear equations to get A, B and C,

From equation 1, `B=-A`

Substitute B in equation 2,


`=>A+C=0`    ---------------------(4)

Add equations 3 and 4,




Plug in the value of A in equation 4,



Plug in the values of A,B and C in the partial fraction template,





Take the constant out,


Apply the sum rule,



Apply the sum rule for the second integral,

`=1/2[int1/(u-1)du-intu/(u^2+1)du-int1/(u^2+1)du]` ------------------(1)

Now let's evaluate each of the above three integrals separately,


Apply integral substitution:`v=u-1`



Use the common integral:`int1/xdx=ln|x|`


Substitute back `v=u-1`

`=ln|u-1|`    -------------------------------------------(2)


Apply integral substitution:`v=u^2+1`



Take the constant out and use standard integral:`int1/xdx=ln|x|`


Substitute back `v=u^2+1`

`=1/2ln|u^2+1|`    ----------------------------------------(3)


Use the common integral:`int1/(x^2+a^2)dx=1/aarctan(x/a)`

`=arctan(u)`  ------------------------------------------(4)

Put the evaluation(2 , 3 and 4) of all the three integrals in (1) ,


Substitute back `u=e^x` and add a constant C to the solution,



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