What is the integral of (1-csc(x))^2 ?

The integral of (1-csc(x))^2 is x-2ln|tan(x/2)|-cot(x)+C

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integral of (1-csc(x))^2

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

x - 2 ln tan (\frac{x}{2}) - cot (x) + C

Solution steps

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

Expand

(1 - csc (x))^{2} : 1 - 2 csc (x) + csc^{2} (x)

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

\int 1 d x = x

\int 2 csc (x) d x = 2 ln tan (\frac{x}{2})

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

= x - 2 ln tan (\frac{x}{2}) - cot (x)

Add a constant to the solution

= x - 2 ln tan (\frac{x}{2}) - cot (x) + C

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

\int 25 t^{- 1} d t

\int (1 - a^{2} x^{2})^{- \frac{3}{2}} d x

\int \frac{25 x ^{2} - 3 ^{2}}{4 x ^{2}} d x

\int \frac{2}{1} d x

What is the integral of (1-csc(x))^2 ?
The integral of (1-csc(x))^2 is x-2ln|tan(x/2)|-cot(x)+C