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 1/(\sqrt[3]{x)+\sqrt[4]{x}}

解

\int \frac{1}{3 x + 4 x} d x

解

12 (\frac{1}{8} x^{\frac{2}{3}} - \frac{1}{7} x^{\frac{7}{12}} + \frac{1}{6} x - \frac{1}{5} x^{\frac{5}{12}} + \frac{1}{4} 3 x - \frac{1}{3} 4 x + \frac{1}{2} 6 x - 12 x + ln 12 x + 1) + C

解答ステップ

\int \frac{1}{3 x + 4 x} d x

u置換積分法を適用する

= \int \frac{12 u ^{8}}{u + 1} d u

定数を除く:

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

= 12 \cdot \int \frac{u ^{8}}{u + 1} d u

長除法

\frac{u ^{8}}{u + 1} : u^{7} - u^{6} + u^{5} - u^{4} + u^{3} - u^{2} + u - 1 + \frac{1}{u + 1}

= 12 \cdot \int u^{7} - u^{6} + u^{5} - u^{4} + u^{3} - u^{2} + u - 1 + \frac{1}{u + 1} 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

= 12 (\int u^{7} d u - \int u^{6} d u + \int u^{5} d u - \int u^{4} d u + \int u^{3} d u - \int u^{2} d u + \int u d u - \int 1 d u + \int \frac{1}{u + 1} d u)

\int u^{7} d u = \frac{u ^{8}}{8}

\int u^{6} d u = \frac{u ^{7}}{7}

\int u^{5} d u = \frac{u ^{6}}{6}

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{3} d u = \frac{u ^{4}}{4}

\int u^{2} d u = \frac{u ^{3}}{3}

\int u d u = \frac{u ^{2}}{2}

\int 1 d u = u

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= 12 (\frac{u ^{8}}{8} - \frac{u ^{7}}{7} + \frac{u ^{6}}{6} - \frac{u ^{5}}{5} + \frac{u ^{4}}{4} - \frac{u ^{3}}{3} + \frac{u ^{2}}{2} - u + ln ∣ u + 1 ∣)

代用を戻す

u = 12 x

= 12 (\frac{( 12 x ) ^{8}}{8} - \frac{( 12 x ) ^{7}}{7} + \frac{( 12 x ) ^{6}}{6} - \frac{( 12 x ) ^{5}}{5} + \frac{( 12 x ) ^{4}}{4} - \frac{( 12 x ) ^{3}}{3} + \frac{( 12 x ) ^{2}}{2} - 12 x + ln 12 x + 1)

簡素化

12 (\frac{( 12 x ) ^{8}}{8} - \frac{( 12 x ) ^{7}}{7} + \frac{( 12 x ) ^{6}}{6} - \frac{( 12 x ) ^{5}}{5} + \frac{( 12 x ) ^{4}}{4} - \frac{( 12 x ) ^{3}}{3} + \frac{( 12 x ) ^{2}}{2} - 12 x + ln 12 x + 1) : 12 (\frac{1}{8} x^{\frac{2}{3}} - \frac{1}{7} x^{\frac{7}{12}} + \frac{1}{6} x - \frac{1}{5} x^{\frac{5}{12}} + \frac{1}{4} 3 x - \frac{1}{3} 4 x + \frac{1}{2} 6 x - 12 x + ln 12 x + 1)

= 12 (\frac{1}{8} x^{\frac{2}{3}} - \frac{1}{7} x^{\frac{7}{12}} + \frac{1}{6} x - \frac{1}{5} x^{\frac{5}{12}} + \frac{1}{4} 3 x - \frac{1}{3} 4 x + \frac{1}{2} 6 x - 12 x + ln 12 x + 1)

定数を解答に追加する

= 12 (\frac{1}{8} x^{\frac{2}{3}} - \frac{1}{7} x^{\frac{7}{12}} + \frac{1}{6} x - \frac{1}{5} x^{\frac{5}{12}} + \frac{1}{4} 3 x - \frac{1}{3} 4 x + \frac{1}{2} 6 x - 12 x + ln 12 x + 1) + C

グラフ

人気の例

limit as x approaches 0-of 6/(5x^{1/3)}

x \to 0 - lim (\frac{6}{5 x ^{\frac{1}{3}}})

integral of (5x^2-2x+3)

\int (5 x^{2} - 2 x + 3) d x

derivative-8/(\sqrt[4]{x)}

d er i v a t i v e - \frac{8}{4 x}

limit as x approaches infinity of 5x+1/x

x \to \infty lim (5 x + \frac{1}{x})

(\partial)/(\partial x)(t^2+4x^3)

\frac{\partial}{\partial x} (t^{2} + 4 x^{3})