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))

integral

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

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

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