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

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

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

\int_{0}^{π} sin (2 x) cos (x) d x

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

\int_{0}^{π} sin (2 x) cos (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

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

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

Apply the Power Rule

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

Simplify

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

Compute the boundaries:

\frac{2}{3}

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

Simplify

= \frac{4}{3}

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