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

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

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

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

\frac{1}{2} sin^{4} (x) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1}{2} sin (2 x) (1 - cos (2 x)) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (2 x) (1 - cos (2 x)) d x

Apply u-substitution

= \frac{1}{2} \cdot \int \frac{u}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int u d u

Apply the Power Rule

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{u ^{2}}{2}

Substitute back

u = 1 - cos (2 x)

= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{( 1 - cos ( 2 x ) ) ^{2}}{2}

Simplify

\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{( 1 - cos ( 2 x ) ) ^{2}}{2} : \frac{1}{2} sin^{4} (x)

= \frac{1}{2} sin^{4} (x)

Add a constant to the solution

= \frac{1}{2} sin^{4} (x) + C

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