What is the integral of (1-sin^3(x))/(sin^2(x)) ?

The integral of (1-sin^3(x))/(sin^2(x)) is -cot(x)+cos(x)+C

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integral of (1-sin^3(x))/(sin^2(x))

\int \frac{1 - sin ^{3} ( x )}{sin ^{2} ( x )} d x

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

Solution steps

\int \frac{1 - sin ^{3} ( x )}{sin ^{2} ( x )} d x

Expand

\frac{1 - sin ^{3} ( x )}{sin ^{2} ( x )} : \frac{1}{sin ^{2} ( x )} - sin (x)

= \int \frac{1}{sin ^{2} ( 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{1}{sin ^{2} ( x )} d x - \int sin (x) d x

\int \frac{1}{sin ^{2} ( x )} d x = - cot (x)

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

= - cot (x) - (- cos (x))

Simplify

= - cot (x) + cos (x)

Add a constant to the solution

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

\frac{d}{d x} (\frac{2 x ^{2} - 3 x + 1}{x})

x^{2} + x y + 2 y^{2} = 28, (2, 3)

t^{2} y^{^{''}} + 7 t y^{^{'}} + 10 y = 0

\int tan (4 x) sec^{5} (4 x) d x

x \to 7 - lim (\frac{1}{x - 7})

What is the integral of (1-sin^3(x))/(sin^2(x)) ?
The integral of (1-sin^3(x))/(sin^2(x)) is -cot(x)+cos(x)+C