Home/Calcolo II/Appendice A: tabella estesa di primitive/A.6. Integrali con 2au−u2\sqrt{2au - u^2}2au−u2ContenutoA.6. Integrali con 2au−u2\sqrt{2au - u^2}2au−u2Condizioni reali usuali: a>0a>0a>0, 0≤u≤2a0\le u\le 2a0≤u≤2a; inoltre, si escludano i valori che annullano un denominatore.FormulaCopia LaTeX∫2au−u2 du=u−a2 2au−u2+a22 arccos(a−ua)+C\int \sqrt{2au - u^2}\,du = \frac{u-a}{2}\,\sqrt{2au - u^2} + \frac{a^2}{2}\,\arccos\left(\frac{a-u}{a}\right) + C∫2au−u2du=2u−a2au−u2+2a2arccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →FormulaCopia LaTeX∫u 2au−u2 du=2u2−au−3a26 2au−u2+a32 arccos(a−ua)+C\int u\,\sqrt{2au - u^2}\,du = \frac{2u^2 - au - 3a^2}{6}\,\sqrt{2au - u^2} + \frac{a^3}{2}\,\arccos\left(\frac{a-u}{a}\right) + C∫u2au−u2du=62u2−au−3a22au−u2+2a3arccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →FormulaCopia LaTeX∫2au−u2u du=2au−u2+a arccos(a−ua)+C\int \frac{\sqrt{2au - u^2}}{u}\,du = \sqrt{2au - u^2} + a\,\arccos\left(\frac{a-u}{a}\right) + C∫u2au−u2du=2au−u2+aarccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →FormulaCopia LaTeX∫2au−u2u2 du=−22au−u2u−arccos(a−ua)+C\int \frac{\sqrt{2au - u^2}}{u^2}\,du = -\frac{2\sqrt{2au - u^2}}{u} - \arccos\left(\frac{a-u}{a}\right) + C∫u22au−u2du=−u22au−u2−arccos(aa−u)+CVedi dettaglio →FormulaCopia LaTeX∫du2au−u2=arccos(a−ua)+C\int \frac{du}{\sqrt{2au - u^2}} = \arccos\left(\frac{a-u}{a}\right) + C∫2au−u2du=arccos(aa−u)+CVedi dettaglio →FormulaCopia LaTeX∫u du2au−u2=−2au−u2+a arccos(a−ua)+C\int \frac{u\,du}{\sqrt{2au - u^2}} = -\sqrt{2au - u^2} + a\,\arccos\left(\frac{a-u}{a}\right) + C∫2au−u2udu=−2au−u2+aarccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →FormulaCopia LaTeX∫u2 du2au−u2=−u+3a2 2au−u2+3a22 arccos(a−ua)+C\int \frac{u^2\,du}{\sqrt{2au - u^2}} = -\frac{u+3a}{2}\,\sqrt{2au - u^2} + \frac{3a^2}{2}\,\arccos\left(\frac{a-u}{a}\right) + C∫2au−u2u2du=−2u+3a2au−u2+23a2arccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →FormulaCopia LaTeX∫duu 2au−u2=−2au−u2au+C\int \frac{du}{u\,\sqrt{2au - u^2}} = -\frac{\sqrt{2au - u^2}}{au} + C∫u2au−u2du=−au2au−u2+CVedi dettaglio →
FormulaCopia LaTeX∫2au−u2 du=u−a2 2au−u2+a22 arccos(a−ua)+C\int \sqrt{2au - u^2}\,du = \frac{u-a}{2}\,\sqrt{2au - u^2} + \frac{a^2}{2}\,\arccos\left(\frac{a-u}{a}\right) + C∫2au−u2du=2u−a2au−u2+2a2arccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →
FormulaCopia LaTeX∫u 2au−u2 du=2u2−au−3a26 2au−u2+a32 arccos(a−ua)+C\int u\,\sqrt{2au - u^2}\,du = \frac{2u^2 - au - 3a^2}{6}\,\sqrt{2au - u^2} + \frac{a^3}{2}\,\arccos\left(\frac{a-u}{a}\right) + C∫u2au−u2du=62u2−au−3a22au−u2+2a3arccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →
FormulaCopia LaTeX∫2au−u2u du=2au−u2+a arccos(a−ua)+C\int \frac{\sqrt{2au - u^2}}{u}\,du = \sqrt{2au - u^2} + a\,\arccos\left(\frac{a-u}{a}\right) + C∫u2au−u2du=2au−u2+aarccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →
FormulaCopia LaTeX∫2au−u2u2 du=−22au−u2u−arccos(a−ua)+C\int \frac{\sqrt{2au - u^2}}{u^2}\,du = -\frac{2\sqrt{2au - u^2}}{u} - \arccos\left(\frac{a-u}{a}\right) + C∫u22au−u2du=−u22au−u2−arccos(aa−u)+CVedi dettaglio →
FormulaCopia LaTeX∫du2au−u2=arccos(a−ua)+C\int \frac{du}{\sqrt{2au - u^2}} = \arccos\left(\frac{a-u}{a}\right) + C∫2au−u2du=arccos(aa−u)+CVedi dettaglio →
FormulaCopia LaTeX∫u du2au−u2=−2au−u2+a arccos(a−ua)+C\int \frac{u\,du}{\sqrt{2au - u^2}} = -\sqrt{2au - u^2} + a\,\arccos\left(\frac{a-u}{a}\right) + C∫2au−u2udu=−2au−u2+aarccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →
FormulaCopia LaTeX∫u2 du2au−u2=−u+3a2 2au−u2+3a22 arccos(a−ua)+C\int \frac{u^2\,du}{\sqrt{2au - u^2}} = -\frac{u+3a}{2}\,\sqrt{2au - u^2} + \frac{3a^2}{2}\,\arccos\left(\frac{a-u}{a}\right) + C∫2au−u2u2du=−2u+3a2au−u2+23a2arccos(aa−u)+Carccos(a−ua)+C\arccos\left(\frac{a-u}{a}\right) + Carccos(aa−u)+CVedi dettaglio →
FormulaCopia LaTeX∫duu 2au−u2=−2au−u2au+C\int \frac{du}{u\,\sqrt{2au - u^2}} = -\frac{\sqrt{2au - u^2}}{au} + C∫u2au−u2du=−au2au−u2+CVedi dettaglio →