integral of x/(x-4)

解答

\int \frac{x}{x - 4} d x

x - 4 + 4 ln ∣ x - 4 ∣ + C

求解步骤

\int \frac{x}{x - 4} d x

使用换元积分法

= \int \frac{u + 4}{u} d u

乘开

\frac{u + 4}{u} : 1 + \frac{4}{u}

= \int 1 + \frac{4}{u} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u + \int \frac{4}{u} d u

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= u + 4 ln ∣ u ∣

u = x - 4

代回

= x - 4 + 4 ln ∣ x - 4 ∣

解答补常数

= x - 4 + 4 ln ∣ x - 4 ∣ + C

y^{^{'''}} + 8 y = 0

\frac{d}{d t} (6 t^{2})

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

\frac{d}{d x} (\frac{x ^{2} - 3 x}{2})

t an g e n t f (x) = 3 x^{2} - x^{3}, at x = 1