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

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

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

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

\frac{2}{3}

Decimal Notation

0.66666 \dots

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int_{1}^{0} - u^{2} d u

= 2 \cdot \int_{1}^{0} - u^{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 2 (- \int_{0}^{1} - u^{2} d u)

Take the constant out:

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

= 2 (- (- \int_{0}^{1} u^{2} d u))

Apply the Power Rule

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

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

Simplify

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

Compute the boundaries:

\frac{1}{3}

= 2 \cdot \frac{1}{3}

Simplify

= \frac{2}{3}

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