integral of (x-2)/(sqrt(x^2-4x+3))

解答

\int \frac{x - 2}{x ^{2} - 4 x + 3} d x

x^{2} - 4 x + 3 + C

求解步骤

\int \frac{x - 2}{x ^{2} - 4 x + 3} d x

使用换元积分法

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

使用幂法则

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

u = x^{2} - 4 x + 3

代回

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

化简

\frac{1}{2} \cdot 2 (x^{2} - 4 x + 3)^{\frac{1}{2}} : x^{2} - 4 x + 3

= x^{2} - 4 x + 3

解答补常数

= x^{2} - 4 x + 3 + C