What is the integral of (-6h+1)/(6h^2-5h+1) ?

The integral of (-6h+1)/(6h^2-5h+1) is ln|3h-1|-2ln|2h-1|+C

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integral of (-6h+1)/(6h^2-5h+1)

\int \frac{- 6 h + 1}{6 h ^{2} - 5 h + 1} d h

ln ∣ 3 h - 1 ∣ - 2 ln ∣ 2 h - 1 ∣ + C

Solution steps

\int \frac{- 6 h + 1}{6 h ^{2} - 5 h + 1} d h

Take the partial fraction of

\frac{- 6 h + 1}{6 h ^{2} - 5 h + 1} : \frac{3}{3 h - 1} - \frac{4}{2 h - 1}

= \int \frac{3}{3 h - 1} - \frac{4}{2 h - 1} d h

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}{3 h - 1} d h - \int \frac{4}{2 h - 1} d h

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

\int \frac{4}{2 h - 1} d h = 2 ln ∣ 2 h - 1 ∣

= ln ∣ 3 h - 1 ∣ - 2 ln ∣ 2 h - 1 ∣

Add a constant to the solution

= ln ∣ 3 h - 1 ∣ - 2 ln ∣ 2 h - 1 ∣ + C

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What is the integral of (-6h+1)/(6h^2-5h+1) ?
The integral of (-6h+1)/(6h^2-5h+1) is ln|3h-1|-2ln|2h-1|+C