What is the integral of [cot(x)-csc(x)]^2 ?

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

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integral of [cot(x)-csc(x)]^2

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

- x - 2 cot (x) + 2 csc (x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

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

= - cot (x) - x - (- 2 csc (x)) - cot (x)

Simplify

= - x - 2 cot (x) + 2 csc (x)

Add a constant to the solution

= - x - 2 cot (x) + 2 csc (x) + C

\int \frac{x + 3}{x ^{2} ( x ^{2} + 9 )} d x

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

\int \frac{1}{1 - cot ( x )} d x

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

\int 1 + x^{\frac{- 2}{3}} d x

What is the integral of [cot(x)-csc(x)]^2 ?
The integral of [cot(x)-csc(x)]^2 is -x-2cot(x)+2csc(x)+C