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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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