0-1-x-2n-arcsinx-dx-

Question Number 156339 by lapache last updated on 10/Oct/21

\int_{0}^{1} \frac{x^{2 n}}{a r c s i n x} d x = \dots ? ? ?

Answered by ArielVyny last updated on 10/Oct/21

t = a r c s i n x \to s i n t = x

\int_{0}^{\frac{π}{2}} \frac{{s i n}^{2 n} t}{t} c o s t d t = \int_{0}^{\frac{π}{2}} \frac{{s i n}^{2 n} t}{t} \sum_{n ⩾ 0} \frac{{(- 1)}^{n} t^{2 n}}{(2 n)!}

\sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{(2 n)!} \int_{0}^{\frac{π}{2}} t^{2 n} {s i n}^{2 n} t d t