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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

= 3 ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{5}{x - 1}

Add a constant to the solution

= 3 ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{5}{x - 1} + C

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