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

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

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

\int (2 - cot (x)) (1 - csc (x)) d x

2 x - 2 ln tan (\frac{x}{2}) - ln ∣ sin (x) ∣ - csc (x) + C

Solution steps

\int (2 - cot (x)) (1 - csc (x)) d x

Rewrite using trig identities

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

Expand

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

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

\int 2 d x = 2 x

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

\int \frac{cos ( x )}{sin ( x )} d x = ln ∣ sin (x) ∣

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

= 2 x - 2 ln tan (\frac{x}{2}) - ln ∣ sin (x) ∣ - csc (x)

Add a constant to the solution

= 2 x - 2 ln tan (\frac{x}{2}) - ln ∣ sin (x) ∣ - csc (x) + C

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