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

The integral of (x-arctan(x))/x is x-1/2 i(\Li_{2}(-ix)-\Li_{2}(ix))+C

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integral of (x-arctan(x))/x

\int \frac{x - arctan ( x )}{x} d x

x - \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x)) + C

Solution steps

\int \frac{x - arctan ( x )}{x} d x

Expand

\frac{x - arctan ( x )}{x} : 1 - \frac{arctan ( x )}{x}

= \int 1 - \frac{arctan ( x )}{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 1 d x - \int \frac{arctan ( x )}{x} d x

\int 1 d x = x

\int \frac{arctan ( x )}{x} d x = \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x))

= x - \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x))

Add a constant to the solution

= x - \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x)) + C

f (x) = lo g_{10} (x + 1)

y^{^{''}} + 2 y^{^{'}} + 10 y = 5 x e^{(- 2 x)}

\frac{2}{x ^{3} + 2}

f (x) = x^{2} - 9

\int x^{2} (x^{3} + 2)^{2} d x

What is the integral of (x-arctan(x))/x ?
The integral of (x-arctan(x))/x is x-1/2 i(\Li_{2}(-ix)-\Li_{2}(ix))+C