| dx | ||
∫ | ||
| √x(1−x) |
| 1 | 1 | ||
t = x− | |||
| 2 | 2 |
| 1 | |
dt = dx | |
| 2 |
| dx | 1/2 dt | 1 | dt | |||||
∫ | =∫ | = | ∫ | = | ||||
| √−(x− 1/2)2 + 1/4 | √−(1/2t)2 + 1/4 | 2 | √1/4(1−t2) |
| dt | ||
= ∫ | = arcsin(t) + C = arcsin(2x−1) + C | |
| √1−t2 |
| π | π | |||
∫ U{dx}{√x(1−x) = arcsin(1) − arcsin(−1) = | − (− | ) = π | ||
| 2 | 2 |
| dx | 2 dx | |||
Funkcja pierwotna = ∫ | = ∫ | = asin(2x−1) | ||
| √1/4−(x−1/2)2 | √1−(2x−1)2 |