What is the integral of sin(x)*sin(2x) ?

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

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integral of sin(x)*sin(2x)

\int sin (x) \cdot sin (2 x) d x

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

Solution steps

\int sin (x) sin (2 x) d x

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

: \int u^{2} d u

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

Apply the Power Rule

: \frac{u ^{3}}{3}

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

Substitute back

u = sin (x)

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

Simplify

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

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

Add a constant to the solution

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

y = x, y = - x + 3, y = 0

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

\frac{d y}{d x} = \frac{y ^{3}}{3 x - 2}

\int \frac{1}{2} x + 3 d x

x \to - 6 lim (\frac{2 x + 12}{x + 6})

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