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

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

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

\int \frac{3 x ^{2} - 1}{( x - 1 ) x ^{2}} d x

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

Solution steps

\int \frac{3 x ^{2} - 1}{( x - 1 ) x ^{2}} d x

Expand

\frac{3 x ^{2} - 1}{( x - 1 ) x ^{2}} : \frac{3 x ^{2}}{( x - 1 ) x ^{2}} - \frac{1}{( x - 1 ) x ^{2}}

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

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{d}{d x} (e^{c o s (x)})

\int x e^{b x} d x

\int sec^{5} (3 x) d x

\frac{d}{d x} (\frac{2 x y}{( 2 - x ^{2} )})

cos (6 y)

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