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

The integral of sin(x)sin((2x)/3) is 3/10 (5sin(x/3)-sin((5x)/3))+C

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

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

\frac{3}{10} (5 sin (\frac{x}{3}) - sin (\frac{5 x}{3})) + C

Solution steps

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

Apply u-substitution

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

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

Rewrite using trig identities

= \frac{3}{2} \cdot \int \frac{1}{2} (- cos (\frac{5 u}{2}) + cos (\frac{u}{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 \frac{1}{2} \cdot \int - cos (\frac{5 u}{2}) + cos (\frac{u}{2}) d u

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} \cdot \frac{1}{2} (- \int cos (\frac{5 u}{2}) d u + \int cos (\frac{u}{2}) d u)

\int cos (\frac{5 u}{2}) d u = \frac{2}{5} sin (\frac{5 u}{2})

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

= \frac{3}{2} \cdot \frac{1}{2} (- \frac{2}{5} sin (\frac{5 u}{2}) + 2 sin (\frac{u}{2}))

Substitute back

u = \frac{2 x}{3}

= \frac{3}{2} \cdot \frac{1}{2} (- \frac{2}{5} sin (\frac{5 \cdot \frac{2 x}{3}}{2}) + 2 sin (\frac{\frac{2 x}{3}}{2}))

Simplify

\frac{3}{2} \cdot \frac{1}{2} (- \frac{2}{5} sin (\frac{5 \cdot \frac{2 x}{3}}{2}) + 2 sin (\frac{\frac{2 x}{3}}{2})) : \frac{3}{10} (5 sin (\frac{x}{3}) - sin (\frac{5 x}{3}))

= \frac{3}{10} (5 sin (\frac{x}{3}) - sin (\frac{5 x}{3}))

Add a constant to the solution

= \frac{3}{10} (5 sin (\frac{x}{3}) - sin (\frac{5 x}{3})) + C

\frac{d}{d x} (e^{a (e^{x} - 1)})

\frac{1}{s}

y = 27 - x^{2}, y = 6 x, y = \frac{2}{5} x

sin (3 x)

\frac{d y}{d x} = 3 x (x - 1)

What is the integral of sin(x)sin((2x)/3) ?
The integral of sin(x)sin((2x)/3) is 3/10 (5sin(x/3)-sin((5x)/3))+C