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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (2 x^{3} y + cos (x^{2}))

\frac{d}{d x} (\frac{x + 2}{x - 2})

\int (\frac{9 + x + x}{x}) d x

\int (2 x + 5) e^{x} d x

y^{^{'}} - 2 x y = e^{x^{2}}

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