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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int sin^{3} (x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int - 1 + u^{2} d u

= \frac{1}{3} \cdot \int - 1 + u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (- \int 1 d u + \int u^{2} d u)

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= \frac{1}{3} (- u + \frac{u ^{3}}{3})

Substitute back

u = cos (x)

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

Add a constant to the solution

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

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