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

The integral of sin^2(x)-sin^3(x) is 1/2 x-1/4 sin(2x)+cos(x)-1/3 cos^3(x)+C

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

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

\frac{1}{2} x - \frac{1}{4} sin (2 x) + cos (x) - \frac{1}{3} cos^{3} (x) + C

Solution steps

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

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

\int sin^{3} (x) d x = - cos (x) + \frac{cos ^{3} ( x )}{3}

= \frac{1}{2} (x - \frac{1}{2} sin (2 x)) - (- cos (x) + \frac{cos ^{3} ( x )}{3})

Simplify

\frac{1}{2} (x - \frac{1}{2} sin (2 x)) - (- cos (x) + \frac{cos ^{3} ( x )}{3}) : \frac{1}{2} x - \frac{1}{4} sin (2 x) + cos (x) - \frac{1}{3} cos^{3} (x)

= \frac{1}{2} x - \frac{1}{4} sin (2 x) + cos (x) - \frac{1}{3} cos^{3} (x)

Add a constant to the solution

= \frac{1}{2} x - \frac{1}{4} sin (2 x) + cos (x) - \frac{1}{3} cos^{3} (x) + C

\frac{450}{s ( s + 3 ) ^{2}}

y = x^{x} + 1

t \to 0 lim (\frac{2}{t} - \frac{2}{t ^{2} + t})

5^{3 x}

(- 2, - 1), (4, 2)

What is the integral of sin^2(x)-sin^3(x) ?
The integral of sin^2(x)-sin^3(x) is 1/2 x-1/4 sin(2x)+cos(x)-1/3 cos^3(x)+C