What is the integral from 0 to pi/3 of sin(2x)sin(x) ?

The integral from 0 to pi/3 of sin(2x)sin(x) is (sqrt(3))/4

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integral from 0 to pi/3 of sin(2x)sin(x)

\int_{0}^{\frac{π}{3}} sin (2 x) sin (x) d x

\frac{3}{4}

Decimal Notation

0.43301 \dots

Solution steps

\int_{0}^{\frac{π}{3}} sin (2 x) sin (x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{3}} 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_{0}^{\frac{π}{3}} sin^{2} (x) cos (x) d x

Apply u-substitution

: \int_{0}^{\frac{3}{2}} u^{2} d u

= 2 \cdot \int_{0}^{\frac{3}{2}} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{0}^{\frac{3}{2}}

= 2 [\frac{u ^{3}}{3}]_{0}^{\frac{3}{2}}

Compute the boundaries:

\frac{3}{8}

= 2 \cdot \frac{3}{8}

Simplify

= \frac{3}{4}

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What is the integral from 0 to pi/3 of sin(2x)sin(x) ?
The integral from 0 to pi/3 of sin(2x)sin(x) is (sqrt(3))/4