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

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

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

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

2 arctan (x) - 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

Solution steps

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

Expand

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

\int \frac{2 x}{x ^{3} + x} d x = 2 arctan (x)

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

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

Add a constant to the solution

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

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

\int \frac{2 x + 5}{2 x + 6} d x

\frac{d}{d x} (\frac{x ^{3}}{3} + \frac{x ^{2}}{2} + e^{- x})

(1 + x^{2})^{- 1}

(- 1.1) (1.6)

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