sin-x-arcsin-x-

Question Number 86671 by M±th+et£s last updated on 30/Mar/20

\int s i n (x) a r c s i n (x)

Answered by Rio Michael last updated on 30/Mar/20

let me give a try .

\int \sin (x) \arcsin (x) d x

using the taylor series expansion for \sin (x) \arcsin (x) centred at 0

\sin (x) \arcsin (x) = x^{2} + \frac{x^{6}}{18} + \frac{x^{8}}{30} + \frac{2669 x^{10}}{113400} + \frac{601 x^{12}}{34020} + \dots

\Rightarrow \int \sin (x) \arcsin (x) d x = \int (x^{2} + \frac{x^{6}}{18} + \frac{x^{8}}{30} + \frac{2669 x^{10}}{113400} + \frac{601 x^{12}}{34020} + \dots) d x

= \frac{x^{3}}{3} + \frac{x^{7}}{126} + \frac{x^{9}}{270} + \frac{2669 x^{11}}{1247400} + \frac{601 x^{13}}{442260} + \dots + k

i still personally think this problem will be cool if we had boundary

conditions like \int_{0}^{1} \sin (x) \arcsin (x) d x .

Commented by M±th+et£s last updated on 30/Mar/20

t h a n k y o u s i r