`int e^x/((e^x-1)(e^x+4)) dx` Use substitution and partial fractions to find the indefinite integral

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Let's apply integral substitution:`u=e^x`



Now create partial fraction template of the integrand,


Multiply the above equation by the denominator,




Equating the coefficients of the like terms,

`A+B=0`    ----------------(1)

`4A-B=1`   ----------------(2)

From equation 1, `A=-B`

Substitute A in equation 2,






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




Apply the sum rule,


Take the constant out,


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


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


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