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

The integral of sin^4(x)sin(2x) is 1/3 sin^6(x)+C

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integral of sin^4(x)sin(2x)

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

\frac{1}{3} sin^{6} (x) + C

Solution steps

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

Rewrite using trig identities

= \int 2 sin^{5} (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^{5} (x) cos (x) d x

Apply u-substitution

= 2 \cdot \int u^{5} d u

Apply the Power Rule

= 2 \cdot \frac{u ^{6}}{6}

Substitute back

u = sin (x)

= 2 \cdot \frac{sin ^{6} ( x )}{6}

Simplify

2 \cdot \frac{sin ^{6} ( x )}{6} : \frac{1}{3} sin^{6} (x)

= \frac{1}{3} sin^{6} (x)

Add a constant to the solution

= \frac{1}{3} sin^{6} (x) + C

What is the integral of sin^4(x)sin(2x) ?
The integral of sin^4(x)sin(2x) is 1/3 sin^6(x)+C