integral of (arctan(5x))/x

解答

\int \frac{arctan ( 5 x )}{x} d x

\frac{1}{2} i (Li_{2} (- i 5 x) - Li_{2} (i 5 x)) + C

求解步骤

\int \frac{arctan ( 5 x )}{x} d x

使用换元积分法

= \int \frac{arctan ( u )}{u} d u

这是非初等函数积分:

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

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

u = 5 x

代回

= \frac{1}{2} i (Li_{2} (- i 5 x) - Li_{2} (i 5 x))

解答补常数

= \frac{1}{2} i (Li_{2} (- i 5 x) - Li_{2} (i 5 x)) + C