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

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

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

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

\frac{2}{5} sin^{5} (x) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int u^{4} d u

= 2 \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= 2 \cdot \frac{u ^{5}}{5}

Substitute back

u = sin (x)

= 2 \cdot \frac{sin ^{5} ( x )}{5}

Simplify

2 \cdot \frac{sin ^{5} ( x )}{5} : \frac{2}{5} sin^{5} (x)

= \frac{2}{5} sin^{5} (x)

Add a constant to the solution

= \frac{2}{5} sin^{5} (x) + C

y = 2 x^{3} + 3 x^{2} - 12 x + 4

x \to - 2 - lim (\frac{1}{( x + 2 ) ^{2}})

x \to 3 lim (\frac{4 x + 2}{x + 4})

x \to 0 lim (3 x - 1)

\int x^{2} sin (6 x) d x

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