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

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

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

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

- 2 csc (x) + 3 cos (x) + C

Solution steps

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

Expand

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

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

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

\int 3 sin (x) d x = - 3 cos (x)

= - 2 csc (x) - (- 3 cos (x))

Simplify

= - 2 csc (x) + 3 cos (x)

Add a constant to the solution

= - 2 csc (x) + 3 cos (x) + C

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