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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

拡張

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

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

総和規則を適用する:

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

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

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

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

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

人気の例

integral from 1 to 7 of (8/x-e^{-x})

\int_{1}^{7} (\frac{8}{x} - e^{- x}) d x

integral from 0 to ln(3) of e^{-x}

\int_{0}^{l n (3)} e^{- x} d x

integral from 1 to 2 of k+(1-k)(x-1)^2

\int_{1}^{2} k + (1 - k) (x - 1)^{2} d x

integral from 0 to 1 of x(1+x^2)^5

\int_{0}^{1} x (1 + x^{2})^{5} d x

integral from 0 to 4 of 1/2 (4cos(5θ))^2

\int_{0}^{4} \frac{1}{2} (4 cos (5 θ))^{2} d θ