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

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

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

\int \frac{x - 1}{x ^{2} + 1} d x

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

Solution steps

\int \frac{x - 1}{x ^{2} + 1} d x

Expand

\frac{x - 1}{x ^{2} + 1} : \frac{x}{x ^{2} + 1} - \frac{1}{x ^{2} + 1}

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{x}{x ^{2} + 1} d x - \int \frac{1}{x ^{2} + 1} d x

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

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

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

Add a constant to the solution

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

\frac{2 u ^{2}}{( u ^{2} + u ) ^{3}}

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

(5 y^{2} - x y) d x + x^{2} d y = 0

\int - \frac{x}{3} d x

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

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