What is the integral from 0 to pi/5 of sin(2x)sin(x) ?

The integral from 0 to pi/5 of sin(2x)sin(x) is ((5-sqrt(5))^{3/2})/(24sqrt(2))

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integral from 0 to pi/5 of sin(2x)sin(x)

\int_{0}^{\frac{π}{5}} sin (2 x) sin (x) d x

\frac{( 5 - 5 ) ^{\frac{3}{2}}}{24 2}

Decimal

0.13538 \dots

Solution steps

\int_{0}^{\frac{π}{5}} sin (2 x) sin (x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{5}} 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}{5}} sin^{2} (x) cos (x) d x

Apply u-substitution

: \int_{0}^{\frac{2 5 - 5}{4}} u^{2} d u

= 2 \cdot \int_{0}^{\frac{2 5 - 5}{4}} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{0}^{\frac{2 5 - 5}{4}}

= 2 [\frac{u ^{3}}{3}]_{0}^{\frac{2 5 - 5}{4}}

\frac{2 5 - 5}{4} = \frac{5 - 5}{2 2}

= 2 [\frac{u ^{3}}{3}]_{0}^{\frac{5 - 5}{2 2}}

Compute the boundaries:

\frac{( 5 - 5 ) ^{\frac{3}{2}}}{48 2}

= 2 \cdot \frac{( 5 - 5 ) ^{\frac{3}{2}}}{48 2}

Simplify

= \frac{( 5 - 5 ) ^{\frac{3}{2}}}{24 2}

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What is the integral from 0 to pi/5 of sin(2x)sin(x) ?
The integral from 0 to pi/5 of sin(2x)sin(x) is ((5-sqrt(5))^{3/2})/(24sqrt(2))