integral of y/((y+2)(5y-1))

解答

\int \frac{y}{( y + 2 ) ( 5 y - 1 )} d y

\frac{2}{11} ln ∣ y + 2 ∣ + \frac{1}{55} ln ∣ 5 y - 1 ∣ + C

求解步骤

\int \frac{y}{( y + 2 ) ( 5 y - 1 )} d y

将

\frac{y}{( y + 2 ) ( 5 y - 1 )}

用部份分式展开:

\frac{2}{11 ( y + 2 )} + \frac{1}{11 ( 5 y - 1 )}

= \int \frac{2}{11 ( y + 2 )} + \frac{1}{11 ( 5 y - 1 )} d y

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}{11 ( y + 2 )} d y + \int \frac{1}{11 ( 5 y - 1 )} d y

\int \frac{2}{11 ( y + 2 )} d y = \frac{2}{11} ln ∣ y + 2 ∣

\int \frac{1}{11 ( 5 y - 1 )} d y = \frac{1}{55} ln ∣ 5 y - 1 ∣

= \frac{2}{11} ln ∣ y + 2 ∣ + \frac{1}{55} ln ∣ 5 y - 1 ∣

解答补常数

= \frac{2}{11} ln ∣ y + 2 ∣ + \frac{1}{55} ln ∣ 5 y - 1 ∣ + C

y^{^{''}} + y^{^{'}} = sin (t)

\int \frac{1}{( x \cdot x ^{2} - 16 )} d x

y = e^{- x} cos (x)

\int \frac{1}{x ^{3} - 4 x ^{2}} d x

\frac{\partial}{\partial x} (\frac{9 x}{7 y + 8 z})