Content

A.6. Integrals with $2 a u - u^{2}$

Usual real conditions: $a > 0$ , $0 \leq u \leq 2 a$ ; also, exclude values that make a denominator zero.

\int 2 a u - u^{2} d u = \frac{u - a}{2} 2 a u - u^{2} + \frac{a ^{2}}{2} arccos (\frac{a - u}{a}) + C

\int u 2 a u - u^{2} d u = \frac{2 u ^{2} - a u - 3 a ^{2}}{6} 2 a u - u^{2} + \frac{a ^{3}}{2} arccos (\frac{a - u}{a}) + C

\int \frac{2 a u - u ^{2}}{u} d u = 2 a u - u^{2} + a arccos (\frac{a - u}{a}) + C

\int \frac{2 a u - u ^{2}}{u ^{2}} d u = - \frac{2 2 a u - u ^{2}}{u} - arccos (\frac{a - u}{a}) + C

\int \frac{d u}{2 a u - u ^{2}} = arccos (\frac{a - u}{a}) + C

\int \frac{u d u}{2 a u - u ^{2}} = - 2 a u - u^{2} + a arccos (\frac{a - u}{a}) + C

\int \frac{u ^{2} d u}{2 a u - u ^{2}} = - \frac{u + 3 a}{2} 2 a u - u^{2} + \frac{3 a ^{2}}{2} arccos (\frac{a - u}{a}) + C

\int \frac{d u}{u 2 a u - u ^{2}} = - \frac{2 a u - u ^{2}}{a u} + C