integral of |2x-6|

解答

\int ∣ 2 x - 6 ∣ d x

\frac{1}{2} ∣ 2 x - 6 ∣ (x - 3) + C

求解步骤

\int ∣ 2 x - 6 ∣ d x

使用换元积分法

= \int ∣ u ∣ \frac{1}{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 ∣ d u

使用积分常见定则:

\int ∣ u ∣ d u = \frac{u ∣ u ∣}{2}

= \frac{1}{2} \cdot \frac{u ∣ u ∣}{2}

u = 2 x - 6

代回

= \frac{1}{2} \cdot \frac{( 2 x - 6 ) ∣ 2 x - 6 ∣}{2}

\frac{( 2 x - 6 ) ∣ 2 x - 6 ∣}{2} = ∣ 2 x - 6 ∣ (x - 3)

= \frac{1}{2} ∣ 2 x - 6 ∣ (x - 3)

解答补常数

= \frac{1}{2} ∣ 2 x - 6 ∣ (x - 3) + C

d er i v a t i v e f (x) = - \frac{5}{x ^{5}}

f (u) = e^{u}

x \to 6 lim (7 x + ∣ x - 6 ∣)

d er i v a t i v e sin (cos (7 x))

\frac{d y}{d x} = - \frac{x}{y e ^{x^{2}}}