What is the integral of (2x^5-x^3-1)/(x^3-4x) ?

The integral of (2x^5-x^3-1)/(x^3-4x) is (2x^3)/3+7x+1/4 ln|x|-57/8 ln|x+2|+55/8 ln|x-2|+C

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integral of (2x^5-x^3-1)/(x^3-4x)

\int \frac{2 x ^{5} - x ^{3} - 1}{x ^{3} - 4 x} d x

\frac{2 x ^{3}}{3} + 7 x + \frac{1}{4} ln ∣ x ∣ - \frac{57}{8} ln ∣ x + 2 ∣ + \frac{55}{8} ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{2 x ^{5} - x ^{3} - 1}{x ^{3} - 4 x} d x

Long division

\frac{2 x ^{5} - x ^{3} - 1}{x ^{3} - 4 x} : 2 x^{2} + 7 + \frac{28 x - 1}{x ^{3} - 4 x}

= \int 2 x^{2} + 7 + \frac{28 x - 1}{x ^{3} - 4 x} 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 2 x^{2} d x + \int 7 d x + \int \frac{28 x - 1}{x ^{3} - 4 x} d x

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

\int 7 d x = 7 x

\int \frac{28 x - 1}{x ^{3} - 4 x} d x = \frac{1}{4} ln ∣ x ∣ - \frac{57}{8} ln ∣ x + 2 ∣ + \frac{55}{8} ln ∣ x - 2 ∣

= \frac{2 x ^{3}}{3} + 7 x + \frac{1}{4} ln ∣ x ∣ - \frac{57}{8} ln ∣ x + 2 ∣ + \frac{55}{8} ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{2 x ^{3}}{3} + 7 x + \frac{1}{4} ln ∣ x ∣ - \frac{57}{8} ln ∣ x + 2 ∣ + \frac{55}{8} ln ∣ x - 2 ∣ + C

x \to 1 + lim (\frac{2 x}{x - 1})

\int_{0}^{2} (2 x^{2} + 1) d x

\int e^{x + 2} d x

\frac{\partial}{\partial m} (m x^{b})

\frac{d}{d x} (x - x^{2})

What is the integral of (2x^5-x^3-1)/(x^3-4x) ?
The integral of (2x^5-x^3-1)/(x^3-4x) is (2x^3)/3+7x+1/4 ln|x|-57/8 ln|x+2|+55/8 ln|x-2|+C