What is the integral of (cot(x))^2 ?

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

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integral of (cot(x))^2

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

- cot (x) - x + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} d x

Expand

\frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} : \frac{1}{sin ^{2} ( x )} - 1

= \int \frac{1}{sin ^{2} ( x )} - 1 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{1}{sin ^{2} ( x )} d x - \int 1 d x

\int \frac{1}{sin ^{2} ( x )} d x = - cot (x)

\int 1 d x = x

= - cot (x) - x

Add a constant to the solution

= - cot (x) - x + C

\int x^{2} \cdot e^{x^{3}} d x

\frac{d}{d x} (- \frac{2 x ^{3}}{y})

\int \frac{y}{( y + 2 ) ( 4 y - 1 )} d y

y = 4.5 x^{2} - 3 x + 2

\frac{\partial}{\partial x} (e^{\frac{x}{2}})