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

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

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

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

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

\int_{0}^{π} cos^{2} (x) sin (x) d x

Apply u-substitution

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

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

= - \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

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

Apply the Power Rule

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

Simplify

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

Compute the boundaries:

\frac{2}{3}

= \frac{2}{3}

\int_{0}^{2} (2 x - 3) (4 x^{2} + 7) d x

\int_{- 0.5}^{0.5} 12 x^{2} - 3 d x

\int_{0}^{1} 9 x e^{x} d x

\int_{- 2}^{0} 4 - x^{2} d x

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

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