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

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

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

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

0

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int_{1}^{- 1} - u^{3} d u

= 2 \cdot \int_{1}^{- 1} - u^{3} d u

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

= 2 (- \int_{- 1}^{1} - u^{3} 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^{3} d u))

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{- 1}^{1}

= 2 (- (- [\frac{u ^{4}}{4}]_{- 1}^{1}))

Simplify

= 2 [\frac{u ^{4}}{4}]_{- 1}^{1}

Compute the boundaries:

0

= 2 \cdot 0

Simplify

= 0

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