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人気のある微分積分 >

integral of (sqrt(6x))/(6+sqrt(x))

解

\int \frac{6 x}{6 + x} d x

解

6 x - 126 x + 726 ln x + 6 - 1086 + C

解答ステップ

\int \frac{6 x}{6 + x} d x

6 x = 6 x

= \int \frac{6 x}{6 + x} d x

定数を除く:

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

= 6 \cdot \int \frac{x}{6 + x} d x

u置換積分法を適用する

= 6 \cdot \int \frac{2 ( u ^{2} - 12 u + 36 )}{u} d u

定数を除く:

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

= 6 \cdot 2 \cdot \int \frac{u ^{2} - 12 u + 36}{u} d u

拡張

\frac{u ^{2} - 12 u + 36}{u} : u - 12 + \frac{36}{u}

= 6 \cdot 2 \cdot \int u - 12 + \frac{36}{u} 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

= 6 \cdot 2 (\int u d u - \int 12 d u + \int \frac{36}{u} d u)

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

\int 12 d u = 12 u

\int \frac{36}{u} d u = 36 ln ∣ u ∣

= 6 \cdot 2 (\frac{u ^{2}}{2} - 12 u + 36 ln ∣ u ∣)

代用を戻す

u = 6 + x

= 6 \cdot 2 (\frac{( 6 + x ) ^{2}}{2} - 12 (6 + x) + 36 ln 6 + x)

簡素化

6 \cdot 2 (\frac{( 6 + x ) ^{2}}{2} - 12 (6 + x) + 36 ln 6 + x) : 6 x - 126 x + 726 ln x + 6 - 1086

= 6 x - 126 x + 726 ln x + 6 - 1086

定数を解答に追加する

= 6 x - 126 x + 726 ln x + 6 - 1086 + C

グラフ

人気の例

derivative of (1+2x/(sqrt(1-x^2)))

\frac{d}{d x} (\frac{1 + 2 x}{1 - x ^{2}})

(\partial)/(\partial x)((2z)/(x+y))

\frac{\partial}{\partial x} (\frac{2 z}{x + y})

derivative of (2x+4x^2/((1+4x)^2))

\frac{d}{d x} (\frac{2 x + 4 x ^{2}}{( 1 + 4 x ) ^{2}})

limit as x approaches-5-of x/(x+5)

x \to - 5 - lim (\frac{x}{x + 5})

slopeintercept (50,43),(60,86)

s l o p e in t erce pt (50, 43), (60, 86)