What is the integral of sin^3(2/3 x)cos^4(2/3 x) ?

The integral of sin^3(2/3 x)cos^4(2/3 x) is 3/2 (-1/5 cos^5((2x)/3)+1/7 cos^7((2x)/3))+C

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integral of sin^3(2/3 x)cos^4(2/3 x)

\int sin^{3} (\frac{2}{3} x) cos^{4} (\frac{2}{3} x) d x

\frac{3}{2} (- \frac{1}{5} cos^{5} (\frac{2 x}{3}) + \frac{1}{7} cos^{7} (\frac{2 x}{3})) + C

Solution steps

\int sin^{3} (\frac{2}{3} x) cos^{4} (\frac{2}{3} x) d x

Apply u-substitution

= \int sin^{3} (u) cos^{4} (u) \frac{3}{2} d u

Take the constant out:

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

= \frac{3}{2} \cdot \int sin^{3} (u) cos^{4} (u) d u

Rewrite using trig identities

= \frac{3}{2} \cdot \int (1 - cos^{2} (u)) sin (u) cos^{4} (u) d u

Apply u-substitution

= \frac{3}{2} \cdot \int - v^{4} (1 - v^{2}) d v

Expand

- v^{4} (1 - v^{2}) : - v^{4} + v^{6}

= \frac{3}{2} \cdot \int - v^{4} + v^{6} 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

= \frac{3}{2} (- \int v^{4} d v + \int v^{6} d v)

\int v^{4} d v = \frac{v ^{5}}{5}

\int v^{6} d v = \frac{v ^{7}}{7}

= \frac{3}{2} (- \frac{v ^{5}}{5} + \frac{v ^{7}}{7})

Substitute back

= \frac{3}{2} (- \frac{cos ^{5} ( \frac{2}{3} x )}{5} + \frac{cos ^{7} ( \frac{2}{3} x )}{7})

Simplify

\frac{3}{2} (- \frac{cos ^{5} ( \frac{2}{3} x )}{5} + \frac{cos ^{7} ( \frac{2}{3} x )}{7}) : \frac{3}{2} (- \frac{1}{5} cos^{5} (\frac{2 x}{3}) + \frac{1}{7} cos^{7} (\frac{2 x}{3}))

= \frac{3}{2} (- \frac{1}{5} cos^{5} (\frac{2 x}{3}) + \frac{1}{7} cos^{7} (\frac{2 x}{3}))

Add a constant to the solution

= \frac{3}{2} (- \frac{1}{5} cos^{5} (\frac{2 x}{3}) + \frac{1}{7} cos^{7} (\frac{2 x}{3})) + C

What is the integral of sin^3(2/3 x)cos^4(2/3 x) ?
The integral of sin^3(2/3 x)cos^4(2/3 x) is 3/2 (-1/5 cos^5((2x)/3)+1/7 cos^7((2x)/3))+C