What is the integral of y/((y+2)(3y-1)) ?

The integral of y/((y+2)(3y-1)) is 2/7 ln|y+2|+1/21 ln|3y-1|+C

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integral of y/((y+2)(3y-1))

\int \frac{y}{( y + 2 ) ( 3 y - 1 )} d y

\frac{2}{7} ln ∣ y + 2 ∣ + \frac{1}{21} ln ∣ 3 y - 1 ∣ + C

Solution steps

\int \frac{y}{( y + 2 ) ( 3 y - 1 )} d y

Take the partial fraction of

\frac{y}{( y + 2 ) ( 3 y - 1 )} : \frac{2}{7 ( y + 2 )} + \frac{1}{7 ( 3 y - 1 )}

= \int \frac{2}{7 ( y + 2 )} + \frac{1}{7 ( 3 y - 1 )} d y

Apply the Sum Rule:

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

= \int \frac{2}{7 ( y + 2 )} d y + \int \frac{1}{7 ( 3 y - 1 )} d y

\int \frac{2}{7 ( y + 2 )} d y = \frac{2}{7} ln ∣ y + 2 ∣

\int \frac{1}{7 ( 3 y - 1 )} d y = \frac{1}{21} ln ∣ 3 y - 1 ∣

= \frac{2}{7} ln ∣ y + 2 ∣ + \frac{1}{21} ln ∣ 3 y - 1 ∣

Add a constant to the solution

= \frac{2}{7} ln ∣ y + 2 ∣ + \frac{1}{21} ln ∣ 3 y - 1 ∣ + C

\frac{d x}{d t} - 0.06 x = 200

2 + \frac{8}{x} + \frac{6}{x ^{2}}

(- sin (x))^{^{'}}

\frac{d y}{d x}, x + y = 5

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

What is the integral of y/((y+2)(3y-1)) ?
The integral of y/((y+2)(3y-1)) is 2/7 ln|y+2|+1/21 ln|3y-1|+C