(Sclipirea Mintii NR 19 - Q35)

  Se $a \in (0, \pi/2]$ and $f:\mathbb{R}\to \mathbb{R}$ is a continuous and odd function, then compute 

 

 $$\int_{-a}^{a}(x^{1936}+x^{2008}+82)\cos^{-1}(\sin(f(x)))dx$$

 

 Solution: (essa solução enviei para referida e confirmada na edição 20)

 

 

$\sin^{-1}y+\cos^{-1}=\frac{\pi}{2}$

$\int_{-a}^{a}(x^{1936}+x^{2008}+82)\cos^{-1}(\sin(f(x)))dx=\int_{-a}^{a}(x^{1936}+x^{2008}+82)\frac{\pi}{2}dx-\int_{-a}^{a}(x^{1936}+x^{2008}+82)f(x)dx$

$\int_{-a}^{a}(x^{1936}+x^{2008}+82)\cos^{-1}(\sin(f(x)))dx=\frac{\pi}{2}\int_{-a}^{a}(x^{1936}+x^{2008}+82)dx=\pi\int_{0}^{a}(x^{1936}+x^{2008}+82)dx$

$\int_{-a}^{a}(x^{1936}+x^{2008}+82)\cos^{-1}(\sin(f(x)))dx= \frac{a^{1937}\pi}{1937}+\ \frac{a^{2009}\pi}{2009}+ 82a\pi$

Ref: https://www.mateinfo.ro/reviste-de-matematica/revista-de-matematica-sclipirea-mintii-issn-1716-3615/revista-sclipirea-mintii/791-revista-de-matematica-sclipirea-mintii-nr-20/file

(pag. 50)