What is the integral of ((tan(x))/(cos(x))-csc^2(x)) ?

The integral of ((tan(x))/(cos(x))-csc^2(x)) is sec(x)+cot(x)+C

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integral of ((tan(x))/(cos(x))-csc^2(x))

\int (\frac{tan ( x )}{cos ( x )} - csc^{2} (x)) d x

sec (x) + cot (x) + C

Solution steps

\int \frac{tan ( x )}{cos ( x )} - csc^{2} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{tan ( x )}{cos ( x )} d x - \int csc^{2} (x) d x

\int \frac{tan ( x )}{cos ( x )} d x = sec (x)

\int csc^{2} (x) d x = - cot (x)

= sec (x) - (- cot (x))

Simplify

= sec (x) + cot (x)

Add a constant to the solution

= sec (x) + cot (x) + C

What is the integral of ((tan(x))/(cos(x))-csc^2(x)) ?
The integral of ((tan(x))/(cos(x))-csc^2(x)) is sec(x)+cot(x)+C