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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int arctan (\frac{x}{4}) d x

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

\int 3 sin (t) d t

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

\int 3 sin^{2} (t) d t

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