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

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Let's apply integral substitution: `u=tan(x)`



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


Multiply the above equation by the denominator,




Equating the coefficients of the like terms,

`A+B=0`    ------------(1)


Plug in the value of A in the equation 1,



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




Apply the sum rule,


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


Substitute back `u=tan(x)`

and add a constant C to the solution,



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