What is the integral of (cos^3(θ))/3 ?

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

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integral of (cos^3(θ))/3

\int \frac{cos ^{3} ( θ )}{3} d θ

\frac{1}{3} (sin (θ) - \frac{sin ^{3} ( θ )}{3}) + C

Solution steps

\int \frac{cos ^{3} ( θ )}{3} d θ

Take the constant out:

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

= \frac{1}{3} \cdot \int cos^{3} (θ) d θ

Rewrite using trig identities

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

Apply u-substitution

= \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 = sin (θ)

= \frac{1}{3} (sin (θ) - \frac{sin ^{3} ( θ )}{3})

Add a constant to the solution

= \frac{1}{3} (sin (θ) - \frac{sin ^{3} ( θ )}{3}) + C

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