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

The integral of (x^5+3)/(x^2-3x+2) is (x^4)/4+x^3+(7x^2)/2+15x-4ln|x-1|+35ln|x-2|+C

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

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

\frac{x ^{4}}{4} + x^{3} + \frac{7 x ^{2}}{2} + 15 x - 4 ln ∣ x - 1 ∣ + 35 ln ∣ x - 2 ∣ + C

Solution steps

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

Long division

\frac{x ^{5} + 3}{x ^{2} - 3 x + 2} : x^{3} + 3 x^{2} + 7 x + 15 + \frac{31 x - 27}{x ^{2} - 3 x + 2}

= \int x^{3} + 3 x^{2} + 7 x + 15 + \frac{31 x - 27}{x ^{2} - 3 x + 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 x^{3} d x + \int 3 x^{2} d x + \int 7 x d x + \int 15 d x + \int \frac{31 x - 27}{x ^{2} - 3 x + 2} d x

\int x^{3} d x = \frac{x ^{4}}{4}

\int 3 x^{2} d x = x^{3}

\int 7 x d x = \frac{7 x ^{2}}{2}

\int 15 d x = 15 x

\int \frac{31 x - 27}{x ^{2} - 3 x + 2} d x = - 4 ln ∣ x - 1 ∣ + 35 ln ∣ x - 2 ∣

= \frac{x ^{4}}{4} + x^{3} + \frac{7 x ^{2}}{2} + 15 x - 4 ln ∣ x - 1 ∣ + 35 ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{x ^{4}}{4} + x^{3} + \frac{7 x ^{2}}{2} + 15 x - 4 ln ∣ x - 1 ∣ + 35 ln ∣ x - 2 ∣ + C

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What is the integral of (x^5+3)/(x^2-3x+2) ?
The integral of (x^5+3)/(x^2-3x+2) is (x^4)/4+x^3+(7x^2)/2+15x-4ln|x-1|+35ln|x-2|+C