What is the integral of (arctan(x)-ln(x)) ?

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

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integral of (arctan(x)-ln(x))

\int (arctan (x) - ln (x)) d x

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

Solution steps

\int arctan (x) - ln (x) 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 arctan (x) d x - \int ln (x) d x

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

\int ln (x) d x = x ln (x) - x

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

Simplify

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

Add a constant to the solution

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

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