What is the integral of (x+10)/(2x^2+5x-3) ?

The integral of (x+10)/(2x^2+5x-3) is 3/2 ln|2x-1|-ln|x+3|+C

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integral of (x+10)/(2x^2+5x-3)

\int \frac{x + 10}{2 x ^{2} + 5 x - 3} d x

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

Solution steps

\int \frac{x + 10}{2 x ^{2} + 5 x - 3} d x

Take the partial fraction of

\frac{x + 10}{2 x ^{2} + 5 x - 3} : \frac{3}{2 x - 1} - \frac{1}{x + 3}

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

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

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

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

Add a constant to the solution

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

What is the integral of (x+10)/(2x^2+5x-3) ?
The integral of (x+10)/(2x^2+5x-3) is 3/2 ln|2x-1|-ln|x+3|+C