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

`int(sec^2(x))/(tan(x)(tan(x)+1))dx`


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


`du=sec^2(x)dx`


`=int1/(u(u+1))du`


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


`1/(u(u+1))=A/u+B/(u+1)`


Multiply the above equation by the denominator,


`1=A(u+1)+B(u)`


`1=Au+A+Bu`


`1=(A+B)u+A`


Equating the coefficients of the like terms,


`A+B=0`    ------------(1)


`A=1`


Plug in the value of A in the equation 1,


`1+B=0`  


`=>B=-1`


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


`1/(u(u+1))=1/u+(-1)/(u+1)`


`=1/u-1/(u+1)`


`int1/(u(u+1))du=int(1/u-1/(u+1))du` 


Apply the sum rule,


`=int1/udu-int1/(u+1)du`


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


`=ln|u|-ln|u+1|`


Substitute back `u=tan(x)`


and add a constant C to the solution,


`=ln|tan(x)|-ln|tan(x)+1|+C`

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