integral of (2x-1)/((x-1)(x-2))

解答

\int \frac{2 x - 1}{( x - 1 ) ( x - 2 )} d x

- ln ∣ x - 1 ∣ + 3 ln ∣ x - 2 ∣ + C

求解步骤

\int \frac{2 x - 1}{( x - 1 ) ( x - 2 )} d x

乘开

\frac{2 x - 1}{( x - 1 ) ( x - 2 )} : \frac{2 x}{( x - 1 ) ( x - 2 )} - \frac{1}{( x - 1 ) ( x - 2 )}

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 \frac{2 x}{( x - 1 ) ( x - 2 )} d x - \int \frac{1}{( x - 1 ) ( x - 2 )} d x

\int \frac{2 x}{( x - 1 ) ( x - 2 )} d x = 2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣)

\int \frac{1}{( x - 1 ) ( x - 2 )} d x = - ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣

= 2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) - (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣)

化简

2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) - (- ln ∣ x - 1 ∣ + ln ∣ x - 2 ∣) : - ln ∣ x - 1 ∣ + 3 ln ∣ x - 2 ∣

= - ln ∣ x - 1 ∣ + 3 ln ∣ x - 2 ∣

解答补常数

= - ln ∣ x - 1 ∣ + 3 ln ∣ x - 2 ∣ + C

l a pl a ce t r an s f or m sin (w)

y^{^{''}} + 4 y^{^{'}} + 5 y = 15 x + e^{(- x)}

y^{^{'}} = 2 x e^{(2 y)}

in v erse l a pl a ce \frac{3}{s \cdot ( s ^{2} + 2 s + 5 )}

\int_{0}^{2 π} cos (x) cos (2 x) d x