integral of 1/(sqrt(1+3x))

解答

\int \frac{1}{1 + 3 x} d x

\frac{2}{3} 1 + 3 x + C

求解步骤

\int \frac{1}{1 + 3 x} d x

使用换元积分法

= \int \frac{1}{3 u} d u

提出常数:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

使用幂法则

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

u = 1 + 3 x

代回

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

化简

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

= \frac{2}{3} 1 + 3 x

解答补常数

= \frac{2}{3} 1 + 3 x + C

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