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

\int \frac{x}{x ^{2} + 4 x + 8} d x

\int_{- 2}^{2} ∣ x ∣ d x

\frac{6}{s ( 2 s ^{2} + 7 s + 3 )}

\frac{( s ^{2} - 1 )}{( s ^{2} + 1 ) ^{2}}

3 x^{2} e^{x} + e^{x} (x^{3} + 3)

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