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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

6 x = 6 x

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

提出常数:

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

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

使用换元积分法

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

提出常数:

\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

乘开

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

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

使用积分加法定则:

\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 ∣)

u = 6 + x

代回

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

化简

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

解答补常数

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

作图

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

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