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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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