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受欢迎的微积分 >

integral of ((3(4u+1)^{1/2}))/4

解答

\int \frac{( 3 ( 4 u + 1 ) ^{\frac{1}{2}} )}{4} d u

解答

\frac{1}{8} (4 u + 1)^{\frac{3}{2}} + C

求解步骤

\int \frac{3 ( 4 u + 1 ) ^{\frac{1}{2}}}{4} d u

提出常数:

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

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

使用换元积分法

= \frac{3}{4} \cdot \int \frac{v ^{\frac{1}{2}}}{4} d v

提出常数:

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

= \frac{3}{4} \cdot \frac{1}{4} \cdot \int v^{\frac{1}{2}} d v

使用幂法则

= \frac{3}{4} \cdot \frac{1}{4} \cdot \frac{2}{3} v^{\frac{3}{2}}

v = 4 u + 1

代回

= \frac{3}{4} \cdot \frac{1}{4} \cdot \frac{2}{3} (4 u + 1)^{\frac{3}{2}}

化简

\frac{3}{4} \cdot \frac{1}{4} \cdot \frac{2}{3} (4 u + 1)^{\frac{3}{2}} : \frac{1}{8} (4 u + 1)^{\frac{3}{2}}

= \frac{1}{8} (4 u + 1)^{\frac{3}{2}}

解答补常数

= \frac{1}{8} (4 u + 1)^{\frac{3}{2}} + C

作图

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