What is the integral of (3-cos(x))/(sin^2(x)) ?

The integral of (3-cos(x))/(sin^2(x)) is -3cot(x)+csc(x)+C

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integral of (3-cos(x))/(sin^2(x))

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

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

Solution steps

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

Expand

\frac{3 - cos ( x )}{sin ^{2} ( x )} : \frac{3}{sin ^{2} ( x )} - \frac{cos ( x )}{sin ^{2} ( x )}

= \int \frac{3}{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{3}{sin ^{2} ( x )} d x - \int \frac{cos ( x )}{sin ^{2} ( x )} d x

\int \frac{3}{sin ^{2} ( x )} d x = - 3 cot (x)

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

= - 3 cot (x) - (- csc (x))

Simplify

= - 3 cot (x) + csc (x)

Add a constant to the solution

= - 3 cot (x) + csc (x) + C

\int_{- 8}^{0} \frac{1}{( x - 8 ) ^{2}} d x

\frac{d y}{d x} = \frac{( y ^{2} + 4 )}{2 x y}

\frac{d y}{d x} = \frac{cos ( 8 x )}{cos ( 4 y )}

\frac{s + 3}{s + 5}

\int e^{i 7 x} cos (3 x) d x

What is the integral of (3-cos(x))/(sin^2(x)) ?
The integral of (3-cos(x))/(sin^2(x)) is -3cot(x)+csc(x)+C