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

The integral of (2x^3+x-1)/(x^2-5x) is x^2+10x+1/5 ln|x|+254/5 ln|x-5|-75+C

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integral of (2x^3+x-1)/(x^2-5x)

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

x^{2} + 10 x + \frac{1}{5} ln ∣ x ∣ + \frac{254}{5} ln ∣ x - 5 ∣ - 75 + C

Solution steps

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

Expand

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

\int \frac{2 x ^{3}}{x ^{2} - 5 x} d x = x^{2} + 10 x + 50 ln ∣ x - 5 ∣ - 75

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

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

= x^{2} + 10 x + 50 ln ∣ x - 5 ∣ - 75 + ln ∣ x - 5 ∣ - (- \frac{1}{5} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 5 ∣)

Simplify

x^{2} + 10 x + 50 ln ∣ x - 5 ∣ - 75 + ln ∣ x - 5 ∣ - (- \frac{1}{5} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 5 ∣) : x^{2} + 10 x + \frac{1}{5} ln ∣ x ∣ + \frac{254}{5} ln ∣ x - 5 ∣ - 75

= x^{2} + 10 x + \frac{1}{5} ln ∣ x ∣ + \frac{254}{5} ln ∣ x - 5 ∣ - 75

Add a constant to the solution

= x^{2} + 10 x + \frac{1}{5} ln ∣ x ∣ + \frac{254}{5} ln ∣ x - 5 ∣ - 75 + C

What is the integral of (2x^3+x-1)/(x^2-5x) ?
The integral of (2x^3+x-1)/(x^2-5x) is x^2+10x+1/5 ln|x|+254/5 ln|x-5|-75+C