integral of x/(x-7)

解答

\int \frac{x}{x - 7} d x

x - 7 + 7 ln ∣ x - 7 ∣ + C

求解步骤

\int \frac{x}{x - 7} d x

使用换元积分法

= \int \frac{u + 7}{u} d u

乘开

\frac{u + 7}{u} : 1 + \frac{7}{u}

= \int 1 + \frac{7}{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{7}{u} d u

\int 1 d u = u

\int \frac{7}{u} d u = 7 ln ∣ u ∣

= u + 7 ln ∣ u ∣

u = x - 7

代回

= x - 7 + 7 ln ∣ x - 7 ∣

解答补常数

= x - 7 + 7 ln ∣ x - 7 ∣ + C

\int 5 sin (6 π) t d t

\int (3 x^{2} - \frac{2}{x ^{3}} + 3 x - 8) d x

d er i v a t i v e \frac{2 x}{x ^{2} + 1}

x^{^{''}} + 3 x^{^{'}} + 2 x = e^{- t}

t an g e n t (3 x + 25)