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

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

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

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

ln ∣ x ∣ + arctan (x) + C

Solution steps

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + arctan (x)

Add a constant to the solution

= ln ∣ x ∣ + arctan (x) + C

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

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

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

\int \frac{1}{- x ^{2} - 4 x + 5} d x

\int \frac{2}{x ( ln ( x ) ) ^{4}} d x

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