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

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

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

integral

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

= - cot (x) + csc (x)

Add a constant to the solution

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

3 t^{2} + 5

e^{\frac{t}{5}}

lo g_{10} (x^{2} + 4 x + 5)

4 sec (x) tan (x) - 2 sec^{2} (x)

\frac{( 1 - 2 v - v ^{2} )}{( 1 + v )}

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