Całka nieoznaczona dotycząca arcusów
: 23 mar 2009, o 20:41
Mam problem z takimi całkami:
\(\displaystyle{ \int \frac{dx}{\sqrt{2+x-x^{2}}}}\)
\(\displaystyle{ \int \frac{dx}{\sqrt{2+x-x^{2}}}}\)
jedziemy na arcsinusa.pascal pisze:Mam problem z takimi całkami:
\(\displaystyle{ \int \frac{dx}{\sqrt{2+x-x^{2}}}}\)
\(\displaystyle{ ...=\int \frac{dx}{ \sqrt{ \frac{5}{4}-(x- \frac{1}{2}) ^{2} } }= \begin{cases} x- \frac{1}{2}= \sqrt{ \frac{5}{4}}t \\ dx= \sqrt{ \frac{5}{4}}dt \end{cases}= \int \frac{ \sqrt{ \frac{5}{4} }dt }{ \sqrt{ \frac{5}{4}- \frac{5}{4}t ^{2} } }=\int \frac{dt}{ \sqrt{1-t ^{2} } }=arcsint+C=arsin( \frac{2x-1}{ \sqrt{5} })+C}\)
Voila