What is the integral of (5-2sin^3(x))/(sin^2(x)) ?

The integral of (5-2sin^3(x))/(sin^2(x)) is -5cot(x)+2cos(x)+C

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integral of (5-2sin^3(x))/(sin^2(x))

\int \frac{5 - 2 sin ^{3} ( x )}{sin ^{2} ( x )} d x

- 5 cot (x) + 2 cos (x) + C

Solution steps

\int \frac{5 - 2 sin ^{3} ( x )}{sin ^{2} ( x )} d x

Expand

\frac{5 - 2 sin ^{3} ( x )}{sin ^{2} ( x )} : \frac{5}{sin ^{2} ( x )} - 2 sin (x)

= \int \frac{5}{sin ^{2} ( x )} - 2 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{5}{sin ^{2} ( x )} d x - \int 2 sin (x) d x

\int \frac{5}{sin ^{2} ( x )} d x = - 5 cot (x)

\int 2 sin (x) d x = - 2 cos (x)

= - 5 cot (x) - (- 2 cos (x))

Simplify

= - 5 cot (x) + 2 cos (x)

Add a constant to the solution

= - 5 cot (x) + 2 cos (x) + C

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What is the integral of (5-2sin^3(x))/(sin^2(x)) ?
The integral of (5-2sin^3(x))/(sin^2(x)) is -5cot(x)+2cos(x)+C