integral from 0 to c of x(c-x)

解答

\int_{0}^{c} x (c - x) d x

\frac{c ^{3}}{2} - \frac{c ^{3}}{3}

求解步骤

\int_{0}^{c} x (c - x) d x

乘开

x (c - x) : c x - x^{2}

= \int_{0}^{c} c x - x^{2} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{c} c x d x - \int_{0}^{c} x^{2} d x

\int_{0}^{c} c x d x = \frac{c ^{3}}{2}

\int_{0}^{c} x^{2} d x = \frac{c ^{3}}{3}

= \frac{c ^{3}}{2} - \frac{c ^{3}}{3}

\int_{- \frac{π}{2}}^{\frac{π}{2}} x d x

\int_{- 1}^{1} [(2 - x^{2}) - (x^{2})] d x

\int_{\frac{π}{3}}^{\frac{π}{2}} cos (π) x d x

\int_{0}^{π} (sin (t) cos (t)) d t

\int_{0}^{1} (2 x + 3)^{10} d x