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

The integral of (x^5+x)/(x-2) is (x^5)/5+(x^4)/2+(4x^3)/3+4x^2+17x+34ln|x-2|-2+C

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

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

\frac{x ^{5}}{5} + \frac{x ^{4}}{2} + \frac{4 x ^{3}}{3} + 4 x^{2} + 17 x + 34 ln ∣ x - 2 ∣ - 2 + C

Solution steps

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

Expand

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

\int \frac{x ^{5}}{x - 2} d x = \frac{x ^{5}}{5} + \frac{x ^{4}}{2} + \frac{4 x ^{3}}{3} + 4 x^{2} + 16 x + 32 ln ∣ x - 2 ∣

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

= \frac{x ^{5}}{5} + \frac{x ^{4}}{2} + \frac{4 x ^{3}}{3} + 4 x^{2} + 16 x + 32 ln ∣ x - 2 ∣ + x - 2 + 2 ln ∣ x - 2 ∣

Simplify

= \frac{x ^{5}}{5} + \frac{x ^{4}}{2} + \frac{4 x ^{3}}{3} + 4 x^{2} + 17 x + 34 ln ∣ x - 2 ∣ - 2

Add a constant to the solution

= \frac{x ^{5}}{5} + \frac{x ^{4}}{2} + \frac{4 x ^{3}}{3} + 4 x^{2} + 17 x + 34 ln ∣ x - 2 ∣ - 2 + C

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What is the integral of (x^5+x)/(x-2) ?
The integral of (x^5+x)/(x-2) is (x^5)/5+(x^4)/2+(4x^3)/3+4x^2+17x+34ln|x-2|-2+C