What is the integral of ((cos(x))/(sin(x)))^2 ?

The integral of ((cos(x))/(sin(x)))^2 is -x-cot(x)+C

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integral of ((cos(x))/(sin(x)))^2

\int (\frac{cos ( x )}{sin ( x )})^{2} d x

- x - cot (x) + C

Solution steps

\int (\frac{cos ( x )}{sin ( x )})^{2} d x

Simplify

(\frac{cos ( x )}{sin ( x )})^{2} : cot^{2} (x)

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

Rewrite using trig identities

= \int - 1 + 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 1 d x + \int csc^{2} (x) d x

\int 1 d x = x

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

= - x - cot (x)

Add a constant to the solution

= - x - cot (x) + C

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