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

The integral of ((2r+1))/(r^2-3r+2) is -3ln|r-1|+5ln|r-2|+C

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integral of ((2r+1))/(r^2-3r+2)

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

- 3 ln ∣ r - 1 ∣ + 5 ln ∣ r - 2 ∣ + C

Solution steps

\int \frac{2 r + 1}{r ^{2} - 3 r + 2} d r

Take the partial fraction of

\frac{2 r + 1}{r ^{2} - 3 r + 2} : - \frac{3}{r - 1} + \frac{5}{r - 2}

= \int - \frac{3}{r - 1} + \frac{5}{r - 2} d r

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{3}{r - 1} d r + \int \frac{5}{r - 2} d r

\int \frac{3}{r - 1} d r = 3 ln ∣ r - 1 ∣

\int \frac{5}{r - 2} d r = 5 ln ∣ r - 2 ∣

= - 3 ln ∣ r - 1 ∣ + 5 ln ∣ r - 2 ∣

Add a constant to the solution

= - 3 ln ∣ r - 1 ∣ + 5 ln ∣ r - 2 ∣ + C

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What is the integral of ((2r+1))/(r^2-3r+2) ?
The integral of ((2r+1))/(r^2-3r+2) is -3ln|r-1|+5ln|r-2|+C