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

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

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

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

3 sin^{2} (\frac{x}{2}) sin (\frac{x}{3}) - \frac{3}{4} (- \frac{3}{4} sin (\frac{4 x}{3}) + \frac{3}{2} sin (\frac{2 x}{3})) + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{3}{2} sin (x) sin (\frac{x}{3}) d x = \frac{3}{4} (- \frac{3}{4} sin (\frac{4 x}{3}) + \frac{3}{2} sin (\frac{2 x}{3}))

= 3 sin^{2} (\frac{x}{2}) sin (\frac{x}{3}) - \frac{3}{4} (- \frac{3}{4} sin (\frac{4 x}{3}) + \frac{3}{2} sin (\frac{2 x}{3}))

Add a constant to the solution

= 3 sin^{2} (\frac{x}{2}) sin (\frac{x}{3}) - \frac{3}{4} (- \frac{3}{4} sin (\frac{4 x}{3}) + \frac{3}{2} sin (\frac{2 x}{3})) + C

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

\frac{d}{d x} (sin (x - 1) x^{3})

\frac{\partial}{\partial z} (\frac{x y}{x ^{2} + y ^{2}})

\frac{3}{x - 6}

y^{^{''}} + 2 y^{^{'}} + 5 y = 0, y (0) = 4, y^{^{'}} (0) = - 3

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