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受欢迎的微积分 >

integral of (2x+3)/(sqrt(x+2))

解答

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

解答

2 (\frac{2}{3} (x + 2)^{\frac{3}{2}} - x + 2) + C

求解步骤

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

使用换元积分法

= \int 2 (2 u^{2} - 1) d u

提出常数:

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

= 2 \cdot \int 2 u^{2} - 1 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

= 2 (\int 2 u^{2} d u - \int 1 d u)

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

\int 1 d u = u

= 2 (\frac{2 u ^{3}}{3} - u)

u = x + 2

代回

= 2 (\frac{2 ( x + 2 ) ^{3}}{3} - x + 2)

化简

2 (\frac{2 ( x + 2 ) ^{3}}{3} - x + 2) : 2 (\frac{2}{3} (x + 2)^{\frac{3}{2}} - x + 2)

= 2 (\frac{2}{3} (x + 2)^{\frac{3}{2}} - x + 2)

解答补常数

= 2 (\frac{2}{3} (x + 2)^{\frac{3}{2}} - x + 2) + C

作图

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