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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

\frac{d}{d z} (\frac{1}{z ^{2}})

ln (1 + e^{- a x})

\int (x + 1) (6 x - 5) d x

\frac{7 x ^{2}}{4 e ^{x} - x}

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

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