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

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

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integral of sin^2(x/2)cos(x/2)

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

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

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int v^{2} d v

= 2 \cdot \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= 2 \cdot \frac{v ^{3}}{3}

Substitute back

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

Simplify

2 \cdot \frac{sin ^{3} ( \frac{x}{2} )}{3} : \frac{2}{3} sin^{3} (\frac{x}{2})

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

Add a constant to the solution

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

f (x) = (6 x + 5) (x^{3} - 2)

x \to + 3 lim (0)

- \frac{y}{x + 5 e}

\int cos^{4} (x) sin^{3} (x) d x

sin (6 x)

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