What is the integral of (2x^4+x^2-x-2)/(x^3(x+1)) ?

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

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integral of (2x^4+x^2-x-2)/(x^3(x+1))

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Simplify

2 (x + 1 - ln ∣ x + 1 ∣) + ln ∣ x ∣ - ln ∣ x + 1 ∣ - (- ln ∣ x ∣ - \frac{1}{x} + ln ∣ x + 1 ∣) - (2 ln ∣ x ∣ + \frac{2}{x} - \frac{1}{x ^{2}} - 2 ln ∣ x + 1 ∣) : 2 x - 2 ln ∣ x + 1 ∣ - \frac{1}{x} + \frac{1}{x ^{2}} + 2

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

Add a constant to the solution

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

\int 2 x sin (x) d x

y^{^{'}} + 8 y = t + e^{- 7 t}

5 x^{3} - 6 cos (x)

f (x) = \frac{2}{x - 8}, (10, 1)

\int csc^{2} (2 x) d x

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