What is the integral of cos^3(1/2 x) ?

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

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integral of cos^3(1/2 x)

\int cos^{3} (\frac{1}{2} x) d x

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

Solution steps

\int cos^{3} (\frac{1}{2} x) d x

Simplify

\frac{1}{2} x : \frac{x}{2}

= \int cos^{3} (\frac{x}{2}) d x

Apply u-substitution

: \int cos^{3} (u) \cdot 2 d u

= \int cos^{3} (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int cos^{3} (u) d u

Rewrite using trig identities

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

Apply u-substitution

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

= 2 \cdot \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

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

\int 1 d v = v

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

= 2 (v - \frac{v ^{3}}{3})

Substitute back

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

Add a constant to the solution

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

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