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

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

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

\int \frac{x - 2}{x ^{2} + x ^{4}} d x

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

Solution steps

\int \frac{x - 2}{x ^{2} + x ^{4}} d x

Apply Integration By Parts

: - 1 - x arctan (x) + \frac{2}{x} + 2 arctan (x) - \int - \frac{1}{x} - arctan (x) d x

= - 1 - x arctan (x) + \frac{2}{x} + 2 arctan (x) - \int - \frac{1}{x} - arctan (x) d x

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

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

Simplify

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

Add a constant to the solution

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

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