Rešite I = ∫ (from 0 to 1) of arcsinxd

$$I = \int _ { 0 } ^ { 1 } \arcsin x d$$

Rešitev za I

$$I=\int _{0}^{1}\arcsin(dx)\mathrm{d}x$$

$$\left(d\leq -\frac{1}{x}\text{ and }d\geq \frac{1}{x}\text{ and }x<0\right)\text{ or }\left(d\geq -\frac{1}{x}\text{ and }d\leq \frac{1}{x}\text{ and }x>0\right)\text{ or }x=0\text{ or }\left(x\neq 0\text{ and }|d|=\frac{1}{|x|}\right)$$

Rešitev za d

$$\left\{\begin{matrix}d=\frac{\sin(I)}{x}\text{, }&\left(x\neq 0\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(I>\pi n_{4}\text{ and }I<\pi n_{4}+\pi \right)\text{ and }|I|\leq \frac{\pi }{2}\right)\text{ or }\left(\exists n_{5}\in \mathrm{Z}\text{ : }I=\frac{\pi n_{5}}{2}\text{, }not(n_{5}<-1)\text{ and }not(n_{5}>1)\text{ and }x\neq 0\right)\text{ or }\left(\exists n_{3}\in \mathrm{Z}\text{ : }I=2\pi n_{3}+\frac{\pi }{2}\text{, }n_{3}=0\text{ and }x\neq 0\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(I>\pi n_{4}\text{ and }I<\pi n_{4}+\pi \right)\right)\text{ or }\left(\exists n_{2}\in \mathrm{Z}\text{ : }I=2\pi n_{2}+\frac{3\pi }{2}\text{, }n_{2}=-1\text{ and }x\neq 0\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(I>\pi n_{4}\text{ and }I<\pi n_{4}+\pi \right)\right)\\d\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }I=\pi n_{1}\text{, }n_{1}=0\text{ and }x=0\end{matrix}\right,$$