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

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

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integral of ((5-x))/((2x^2+x-1))

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

\frac{3}{2} ln ∣ 2 x - 1 ∣ - 2 ln ∣ x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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}{2 x - 1} d x - \int \frac{2}{x + 1} d x

\int \frac{3}{2 x - 1} d x = \frac{3}{2} ln ∣ 2 x - 1 ∣

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

= \frac{3}{2} ln ∣ 2 x - 1 ∣ - 2 ln ∣ x + 1 ∣

Add a constant to the solution

= \frac{3}{2} ln ∣ 2 x - 1 ∣ - 2 ln ∣ x + 1 ∣ + C

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