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 2x^3(x^4-4)^4

解

\int 2 x^{3} (x^{4} - 4)^{4} d x

解

\frac{1}{10} x^{4} (x^{16} - 20 x^{12} + 160 x^{8} - 640 x^{4} + 1280) + C

解答ステップ

\int 2 x^{3} (x^{4} - 4)^{4} d x

定数を除く:

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

= 2 \cdot \int x^{3} (x^{4} - 4)^{4} d x

u置換積分法を適用する

= 2 \cdot \int \frac{u ^{\frac{1}{3}} ( u ^{\frac{4}{3}} - 4 ) ^{4}}{3} d u

定数を除く:

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

= 2 \cdot \frac{1}{3} \cdot \int u^{\frac{1}{3}} (u^{\frac{4}{3}} - 4)^{4} d u

拡張

u^{\frac{1}{3}} (u^{\frac{4}{3}} - 4)^{4} : u^{\frac{17}{3}} - 16 u^{\frac{13}{3}} + 96 u^{3} - 256 u^{\frac{5}{3}} + 256 u^{\frac{1}{3}}

= 2 \cdot \frac{1}{3} \cdot \int u^{\frac{17}{3}} - 16 u^{\frac{13}{3}} + 96 u^{3} - 256 u^{\frac{5}{3}} + 256 u^{\frac{1}{3}} 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

= 2 \cdot \frac{1}{3} (\int u^{\frac{17}{3}} d u - \int 16 u^{\frac{13}{3}} d u + \int 96 u^{3} d u - \int 256 u^{\frac{5}{3}} d u + \int 256 u^{\frac{1}{3}} d u)

\int u^{\frac{17}{3}} d u = \frac{3}{20} u^{\frac{20}{3}}

\int 16 u^{\frac{13}{3}} d u = 3 u^{\frac{16}{3}}

\int 96 u^{3} d u = 24 u^{4}

\int 256 u^{\frac{5}{3}} d u = 96 u^{\frac{8}{3}}

\int 256 u^{\frac{1}{3}} d u = 192 u^{\frac{4}{3}}

= 2 \cdot \frac{1}{3} (\frac{3}{20} u^{\frac{20}{3}} - 3 u^{\frac{16}{3}} + 24 u^{4} - 96 u^{\frac{8}{3}} + 192 u^{\frac{4}{3}})

代用を戻す

u = x^{3}

= 2 \cdot \frac{1}{3} (\frac{3}{20} (x^{3})^{\frac{20}{3}} - 3 (x^{3})^{\frac{16}{3}} + 24 (x^{3})^{4} - 96 (x^{3})^{\frac{8}{3}} + 192 (x^{3})^{\frac{4}{3}})

簡素化

2 \cdot \frac{1}{3} (\frac{3}{20} (x^{3})^{\frac{20}{3}} - 3 (x^{3})^{\frac{16}{3}} + 24 (x^{3})^{4} - 96 (x^{3})^{\frac{8}{3}} + 192 (x^{3})^{\frac{4}{3}}) : \frac{1}{10} x^{4} (x^{16} - 20 x^{12} + 160 x^{8} - 640 x^{4} + 1280)

= \frac{1}{10} x^{4} (x^{16} - 20 x^{12} + 160 x^{8} - 640 x^{4} + 1280)

定数を解答に追加する

= \frac{1}{10} x^{4} (x^{16} - 20 x^{12} + 160 x^{8} - 640 x^{4} + 1280) + C

グラフ

人気の例

integral of sin(a)xcos(a)x

\int sin (a) x cos (a) x d x

integral of (e^t-e^{-t})

\int (e^{t} - e^{- t}) d t

integral of 10sec(x)tan(x)

\int 10 sec (x) tan (x) d x

integral of-3e^{cos(x)}sin(x)

\int - 3 e^{c o s (x)} sin (x) d x

integral of (1+x^2)^3

\int (1 + x^{2})^{3} d x