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

The integral of sin^3(x)-sin^5(x) is 1/3 cos^3(x)-2/3 cos^3(x)+1/5 cos^5(x)+C

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

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

\frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x) + C

Solution steps

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

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

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

= - cos (x) + \frac{cos ^{3} ( x )}{3} - (- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5})

Simplify

- cos (x) + \frac{cos ^{3} ( x )}{3} - (- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}) : \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x)

= \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x)

Add a constant to the solution

= \frac{1}{3} cos^{3} (x) - \frac{2}{3} cos^{3} (x) + \frac{1}{5} cos^{5} (x) + C

What is the integral of sin^3(x)-sin^5(x) ?
The integral of sin^3(x)-sin^5(x) is 1/3 cos^3(x)-2/3 cos^3(x)+1/5 cos^5(x)+C