What is the integral of (x^5+1)/(x^3(x+2)) ?

The integral of (x^5+1)/(x^3(x+2)) is (x^2)/2-2x+1/8 ln|x|+1/(4x)-1/(4x^2)+31/8 ln|x+2|+C

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integral of (x^5+1)/(x^3(x+2))

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

\frac{x ^{2}}{2} - 2 x + \frac{1}{8} ln ∣ x ∣ + \frac{1}{4 x} - \frac{1}{4 x ^{2}} + \frac{31}{8} ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x ^{5} + 1}{x ^{3} ( x + 2 )} : x - 2 + \frac{1}{8 x} - \frac{1}{4 x ^{2}} + \frac{1}{2 x ^{3}} + \frac{31}{8 ( x + 2 )}

= \int x - 2 + \frac{1}{8 x} - \frac{1}{4 x ^{2}} + \frac{1}{2 x ^{3}} + \frac{31}{8 ( 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 d x - \int 2 d x + \int \frac{1}{8 x} d x - \int \frac{1}{4 x ^{2}} d x + \int \frac{1}{2 x ^{3}} d x + \int \frac{31}{8 ( x + 2 )} d x

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

\int 2 d x = 2 x

\int \frac{1}{8 x} d x = \frac{1}{8} ln ∣ x ∣

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

\int \frac{1}{2 x ^{3}} d x = - \frac{1}{4 x ^{2}}

\int \frac{31}{8 ( x + 2 )} d x = \frac{31}{8} ln ∣ x + 2 ∣

= \frac{x ^{2}}{2} - 2 x + \frac{1}{8} ln ∣ x ∣ - (- \frac{1}{4 x}) - \frac{1}{4 x ^{2}} + \frac{31}{8} ln ∣ x + 2 ∣

Simplify

= \frac{x ^{2}}{2} - 2 x + \frac{1}{8} ln ∣ x ∣ + \frac{1}{4 x} - \frac{1}{4 x ^{2}} + \frac{31}{8} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - 2 x + \frac{1}{8} ln ∣ x ∣ + \frac{1}{4 x} - \frac{1}{4 x ^{2}} + \frac{31}{8} ln ∣ x + 2 ∣ + C

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What is the integral of (x^5+1)/(x^3(x+2)) ?
The integral of (x^5+1)/(x^3(x+2)) is (x^2)/2-2x+1/8 ln|x|+1/(4x)-1/(4x^2)+31/8 ln|x+2|+C