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人気のある微分積分 >

integral from 0 to pi/2 of-5cos^5(x)

解

\int_{0}^{\frac{π}{2}} - 5 cos^{5} (x) d x

解

- \frac{8}{3}

+1

十進法表記

- 2.66666 \dots

解答ステップ

\int_{0}^{\frac{π}{2}} - 5 cos^{5} (x) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 5 \cdot \int_{0}^{\frac{π}{2}} cos^{5} (x) d x

三角関数の公式を使用して書き換える

= - 5 \cdot \int_{0}^{\frac{π}{2}} (1 - sin^{2} (x))^{2} cos (x) d x

u置換積分法を適用する

= - 5 \cdot \int_{0}^{1} (1 - u^{2})^{2} d u

拡張

(1 - u^{2})^{2} : 1 - 2 u^{2} + u^{4}

= - 5 \cdot \int_{0}^{1} 1 - 2 u^{2} + u^{4} d u

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

= - 5 (\int_{0}^{1} 1 d u - \int_{0}^{1} 2 u^{2} d u + \int_{0}^{1} u^{4} d u)

\int_{0}^{1} 1 d u = 1

\int_{0}^{1} 2 u^{2} d u = \frac{2}{3}

\int_{0}^{1} u^{4} d u = \frac{1}{5}

= - 5 (1 - \frac{2}{3} + \frac{1}{5})

簡素化

- 5 (1 - \frac{2}{3} + \frac{1}{5}) : - \frac{8}{3}

= - \frac{8}{3}

グラフ

人気の例

derivative of sqrt(3-x^2)

\frac{d}{d x} (3 - x^{2})

sum from n=3 to infinity of 2/(n(n+3))

n = 3 \sum \infty \frac{2}{n ( n + 3 )}

(dh)/(dt)=4-16t^2,\quad h(1)=0

\frac{d h}{d t} = 4 - 16 t^{2}, h (1) = 0

limit as x approaches 1+of 8/(x^3-1)

x \to 1 + lim (\frac{8}{x ^{3} - 1})

tangent f(x)=(x^3-9x)^{10},(-3,0)

t an g e n t f (x) = (x^{3} - 9 x)^{10}, (- 3, 0)