What is the integral of (cos(θ))^3 ?

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

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integral of (cos(θ))^3

\int (cos (θ))^{3} d θ

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

Solution steps

\int (cos (θ))^{3} d θ

Simplify

= \int cos^{3} (θ) d θ

Rewrite using trig identities

= \int (1 - sin^{2} (θ)) cos (θ) d θ

Apply u-substitution

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

= \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

= \int 1 d u - \int u^{2} d u

\int 1 d u = u

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

= u - \frac{u ^{3}}{3}

Substitute back

u = sin (θ)

= sin (θ) - \frac{sin ^{3} ( θ )}{3}

Add a constant to the solution

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

\int_{0}^{\frac{π}{3}} x sin (x + y) d y

\int sin (x) + 3 cos (x) d x

\frac{d}{d x} ((1 + 2 x))

x \to 4 - lim (\frac{3}{x ^{2} - 16})

y = tan (2) (x)

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