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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

= - cos (x) - cot (x)

Add a constant to the solution

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

\int 6 sin (x) - 3 cos (x) d x

\int 18 sin^{2} (x) d x

\int 5 x^{3} + e^{x} - cos (x) d x

\int \frac{x ^{2} + 4 x + 2}{x ^{2} + 1} d x

\int \frac{1}{( x x ^{2} - 9 )} d x

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