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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

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

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

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

Add a constant to the solution

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

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