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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x+1)/(x^2(x^2+1))

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

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

Solution steps

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

Apply Integration By Parts

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

Simplify

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

Add a constant to the solution

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

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