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

The integral from 0 to pi/4 of sin(2x)sin(x) is 1/(3sqrt(2))

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integral from 0 to pi/4 of sin(2x)sin(x)

\int_{0}^{\frac{π}{4}} sin (2 x) sin (x) d x

\frac{1}{3 2}

Decimal

0.23570 \dots

Solution steps

\int_{0}^{\frac{π}{4}} sin (2 x) sin (x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{4}} 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}{4}} sin^{2} (x) cos (x) d x

Apply u-substitution

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

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

Apply the Power Rule

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

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

\frac{2}{2} = \frac{1}{2}

= 2 [\frac{u ^{3}}{3}]_{0}^{\frac{1}{2}}

Compute the boundaries:

\frac{1}{6 2}

= 2 \cdot \frac{1}{6 2}

Simplify

= \frac{1}{3 2}

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\int_{0}^{6} 2 x x^{2} + 64 d x

\int_{0}^{\infty} \frac{1}{x ^{p}} d x

What is the integral from 0 to pi/4 of sin(2x)sin(x) ?
The integral from 0 to pi/4 of sin(2x)sin(x) is 1/(3sqrt(2))