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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

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