`int (sec^2x)/(tan^2x+5tanx+6) dx` Use substitution and partial fractions to find the indefinite integral

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



Now we have to write down integrand as sum of partial fraction function, but first we will have to factor the denominator,




Now let's create partial fraction template,


Multiply the above equation by the denominator,




Equating the coefficients of the like terms,

`A+B=0`      -----------------(1)

`3A+2B=1`  -----------------(2)

From equation 1:`A=-B`

Substitute A in equation 2,




Plug in the values 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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