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人気のある微分積分 >

integral from 0 to 1 of 3(1+sqrt(x))^5

解

\int_{0}^{1} 3 (1 + x)^{5} d x

解

\frac{321}{7}

+1

十進法表記

45.85714 \dots

解答ステップ

\int_{0}^{1} 3 (1 + x)^{5} d x

定数を除く:

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

= 3 \cdot \int_{0}^{1} (1 + x)^{5} d x

拡張

(1 + x)^{5} : 1 + 5 x + 10 x + 10 x^{\frac{3}{2}} + 5 x^{2} + x^{\frac{5}{2}}

= 3 \cdot \int_{0}^{1} 1 + 5 x + 10 x + 10 x^{\frac{3}{2}} + 5 x^{2} + x^{\frac{5}{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

= 3 (\int_{0}^{1} 1 d x + \int_{0}^{1} 5 x d x + \int_{0}^{1} 10 x d x + \int_{0}^{1} 10 x^{\frac{3}{2}} d x + \int_{0}^{1} 5 x^{2} d x + \int_{0}^{1} x^{\frac{5}{2}} d x)

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

\int_{0}^{1} 5 x d x = \frac{10}{3}

\int_{0}^{1} 10 x d x = 5

\int_{0}^{1} 10 x^{\frac{3}{2}} d x = 4

\int_{0}^{1} 5 x^{2} d x = \frac{5}{3}

\int_{0}^{1} x^{\frac{5}{2}} d x = \frac{2}{7}

= 3 (1 + \frac{10}{3} + 5 + 4 + \frac{5}{3} + \frac{2}{7})

簡素化

3 (1 + \frac{10}{3} + 5 + 4 + \frac{5}{3} + \frac{2}{7}) : \frac{321}{7}

= \frac{321}{7}

グラフ

人気の例

integral from 0 to 1 of (2x-3)

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

derivative f(x)=(x^3+2x+7)/(x^2-5)

d er i v a t i v e f (x) = \frac{x ^{3} + 2 x + 7}{x ^{2} - 5}

integral from 1 to 1.5 of 2/5 (x+1)

\int_{1}^{1.5} \frac{2}{5} (x + 1) d x

derivative f(t)=6t^3-54t^2+70

d er i v a t i v e f (t) = 6 t^{3} - 54 t^{2} + 70

derivative of 1/(\sqrt[3]{x^3+1})

\frac{d}{d x} (\frac{1}{3 x ^{3} + 1})