`int (tan^2x)/secx dx` Find the indefinite integral

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

Transform the numerator of the integral by using the following trigonometric identity:

`tan^2(x)=sec^2(x)-1`

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

`=int((sec^2(x))/sec(x)-1/sec(x))dx`

`=int(sec(x)-cos(x))dx`

Apply the sum rule:

`=intsec(x)dx-intcos(x)dx`

use the following common integrals:

`intsec(x)dx=ln|sec(x)+tan(x)|+C`

and `intcos(x)dx=sin(x)+C`

`=ln|sec(x)+tan(x)|-sin(x)+C`

Comments

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