What is the integral of ((x^2+5x+4))/(x^2-2x+1) ?

The integral of ((x^2+5x+4))/(x^2-2x+1) is (x^3)/(x^2-2x+1)+7ln|x-1|-(13)/(x-1)-1/((x-1)^2)+C

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integral of ((x^2+5x+4))/(x^2-2x+1)

\int \frac{( x ^{2} + 5 x + 4 )}{x ^{2} - 2 x + 1} d x

\frac{x ^{3}}{x ^{2} - 2 x + 1} + 7 ln ∣ x - 1 ∣ - \frac{13}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + C

Solution steps

\int \frac{x ^{2} + 5 x + 4}{x ^{2} - 2 x + 1} d x

Expand

\frac{x ^{2} + 5 x + 4}{x ^{2} - 2 x + 1} : \frac{x ^{2}}{x ^{2} - 2 x + 1} + \frac{5 x}{x ^{2} - 2 x + 1} + \frac{4}{x ^{2} - 2 x + 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{x ^{2}}{x ^{2} - 2 x + 1} d x + \int \frac{5 x}{x ^{2} - 2 x + 1} d x + \int \frac{4}{x ^{2} - 2 x + 1} d x

\int \frac{x ^{2}}{x ^{2} - 2 x + 1} d x = \frac{x ^{3}}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{4}{x - 1} - \frac{1}{( x - 1 ) ^{2}}

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

\int \frac{4}{x ^{2} - 2 x + 1} d x = - \frac{4}{x - 1}

= \frac{x ^{3}}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{4}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + 5 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - \frac{4}{x - 1}

Simplify

\frac{x ^{3}}{x ^{2} - 2 x + 1} + 2 ln ∣ x - 1 ∣ - \frac{4}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + 5 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - \frac{4}{x - 1} : \frac{x ^{3}}{x ^{2} - 2 x + 1} + 7 ln ∣ x - 1 ∣ - \frac{13}{x - 1} - \frac{1}{( x - 1 ) ^{2}}

= \frac{x ^{3}}{x ^{2} - 2 x + 1} + 7 ln ∣ x - 1 ∣ - \frac{13}{x - 1} - \frac{1}{( x - 1 ) ^{2}}

Add a constant to the solution

= \frac{x ^{3}}{x ^{2} - 2 x + 1} + 7 ln ∣ x - 1 ∣ - \frac{13}{x - 1} - \frac{1}{( x - 1 ) ^{2}} + C

What is the integral of ((x^2+5x+4))/(x^2-2x+1) ?
The integral of ((x^2+5x+4))/(x^2-2x+1) is (x^3)/(x^2-2x+1)+7ln|x-1|-(13)/(x-1)-1/((x-1)^2)+C