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

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

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integral of sin^3(x/2)

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

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

Solution steps

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

Apply u-substitution

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

= \int sin^{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 sin^{3} (u) d u

Rewrite using trig identities

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

Add a constant to the solution

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

θ \to \infty lim (e^{- 3 θ} cos (θ))

h \to 0 lim (\frac{( a ^{u + h} - a ^{u} )}{h})

\int e^{- a x} sin (b x) d x

\int (e^{3 x} + 5 e^{- x}) d x

\frac{d}{d x} (c_{1} cos (x) + c_{2} sin (x))

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