ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of (x^3)/(\sqrt[3]{x^2+5)}

解

\int \frac{x ^{3}}{3 x ^{2} + 5} d x

解

\frac{1}{20} (6 (x^{2} + 5)^{\frac{5}{3}} - 75 (x^{2} + 5)^{\frac{2}{3}}) + C

解答ステップ

\int \frac{x ^{3}}{3 x ^{2} + 5} d x

u置換積分法を適用する

= \int \frac{u ^{\frac{1}{3}}}{3 3 u ^{\frac{2}{3}} + 5} d u

定数を除く:

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

= \frac{1}{3} \cdot \int \frac{u ^{\frac{1}{3}}}{3 u ^{\frac{2}{3}} + 5} d u

u置換積分法を適用する

= \frac{1}{3} \cdot \int \frac{3 ( v - 5 )}{2 3 v} d v

定数を除く:

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int \frac{v - 5}{3 v} d v

拡張

\frac{v - 5}{3 v} : v^{\frac{2}{3}} - \frac{5}{3 v}

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int v^{\frac{2}{3}} - \frac{5}{3 v} d v

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

= \frac{1}{3} \cdot \frac{3}{2} (\int v^{\frac{2}{3}} d v - \int \frac{5}{3 v} d v)

\int v^{\frac{2}{3}} d v = \frac{3}{5} v^{\frac{5}{3}}

\int \frac{5}{3 v} d v = \frac{15}{2} v^{\frac{2}{3}}

= \frac{1}{3} \cdot \frac{3}{2} (\frac{3}{5} v^{\frac{5}{3}} - \frac{15}{2} v^{\frac{2}{3}})

代用を戻す

= \frac{1}{3} \cdot \frac{3}{2} (\frac{3}{5} ((x^{3})^{\frac{2}{3}} + 5)^{\frac{5}{3}} - \frac{15}{2} ((x^{3})^{\frac{2}{3}} + 5)^{\frac{2}{3}})

簡素化

\frac{1}{3} \cdot \frac{3}{2} (\frac{3}{5} ((x^{3})^{\frac{2}{3}} + 5)^{\frac{5}{3}} - \frac{15}{2} ((x^{3})^{\frac{2}{3}} + 5)^{\frac{2}{3}}) : \frac{1}{20} (6 (x^{2} + 5)^{\frac{5}{3}} - 75 (x^{2} + 5)^{\frac{2}{3}})

= \frac{1}{20} (6 (x^{2} + 5)^{\frac{5}{3}} - 75 (x^{2} + 5)^{\frac{2}{3}})

定数を解答に追加する

= \frac{1}{20} (6 (x^{2} + 5)^{\frac{5}{3}} - 75 (x^{2} + 5)^{\frac{2}{3}}) + C

グラフ

人気の例

integral from 0 to pi/4 of cos^7(x)

\int_{0}^{\frac{π}{4}} cos^{7} (x) d x

(\partial)/(\partial x)(y^2-4y+xy)

\frac{\partial}{\partial x} (y^{2} - 4 y + x y)

derivative (x^3)/8

d er i v a t i v e \frac{x ^{3}}{8}

integral of cos^4(6x)

\int cos^{4} (6 x) d x

derivative of 1/e e^x

\frac{d}{d x} (\frac{1}{e} e^{x})