What is the integral of cos^3(u) ?

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

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integral of cos^3(u)

\int cos^{3} (u) d u

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

Solution steps

\int cos^{3} (u) d u

Rewrite using trig identities

= \int (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

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

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

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 v - \int v^{2} d v

\int 1 d v = v

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

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

Substitute back

v = sin (u)

= sin (u) - \frac{sin ^{3} ( u )}{3}

Add a constant to the solution

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

x \to π lim (cos (3 x))

\frac{d}{d x} (2 x 64 - x^{2})

x \to - 1 lim (- 1^{3})

\frac{d}{d x} (x^{2} - x)

\int tan^{2} (x) d tan (x) d x

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