What is the integral of ((1-cos(x))^2)/(sin^2(x)) ?

The integral of ((1-cos(x))^2)/(sin^2(x)) is -x-2cot(x)+2csc(x)+C

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integral of ((1-cos(x))^2)/(sin^2(x))

integral

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

in t e g r a l 3^{- x}

in t e g r a l \frac{arctan ( x )}{x ^{2}}

in t e g r a l \frac{1}{( x - 2 ) ^{3} ( x + 4 ) ^{4}}

in t e g r a l (1 + x tan (x) - sin (x)) sec^{2} (x)

in t e g r a l cos^{2} (x) - sin^{2} (x)

What is the integral of ((1-cos(x))^2)/(sin^2(x)) ?
The integral of ((1-cos(x))^2)/(sin^2(x)) is -x-2cot(x)+2csc(x)+C