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

x d y - y d x = 0

\frac{- 2 x}{( x ^{2} + 1 )}

\int \frac{3 x}{4} d x

x \to \infty lim (x - 0)

\int_{3}^{\infty} \frac{1}{x ^{4}} d x

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