What is the integral of (acos(x)+b)/(sin^2(x)) ?

The integral of (acos(x)+b)/(sin^2(x)) is -acsc(x)-bcot(x)+C

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integral of (acos(x)+b)/(sin^2(x))

\int \frac{a cos ( x ) + b}{sin ^{2} ( x )} d x

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

Solution steps

\int \frac{a cos ( x ) + b}{sin ^{2} ( x )} d x

Expand

\frac{a cos ( x ) + b}{sin ^{2} ( x )} : \frac{a cos ( x )}{sin ^{2} ( x )} + \frac{b}{sin ^{2} ( x )}

= \int \frac{a cos ( x )}{sin ^{2} ( x )} + \frac{b}{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{a cos ( x )}{sin ^{2} ( x )} d x + \int \frac{b}{sin ^{2} ( x )} d x

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

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

= - a csc (x) - b cot (x)

Add a constant to the solution

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

What is the integral of (acos(x)+b)/(sin^2(x)) ?
The integral of (acos(x)+b)/(sin^2(x)) is -acsc(x)-bcot(x)+C