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受欢迎的微积分 >

integral of (2x+3)^{1/2}

解答

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

解答

\frac{1}{3} (2 x + 3)^{\frac{3}{2}} + C

求解步骤

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

使用换元积分法

= \int \frac{u ^{\frac{1}{2}}}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int u^{\frac{1}{2}} d u

使用幂法则

= \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

u = 2 x + 3

代回

= \frac{1}{2} \cdot \frac{2}{3} (2 x + 3)^{\frac{3}{2}}

化简

\frac{1}{2} \cdot \frac{2}{3} (2 x + 3)^{\frac{3}{2}} : \frac{1}{3} (2 x + 3)^{\frac{3}{2}}

= \frac{1}{3} (2 x + 3)^{\frac{3}{2}}

解答补常数

= \frac{1}{3} (2 x + 3)^{\frac{3}{2}} + C

作图

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