| (3x2 − 5x + 2)dx | ||
∫ ( | ) | |
| x3 − 2x2 + 3x − 6 |
| 3x2−5x+2 | ||
∫ | dx= | |
| x3−2x2+3x−6 |
| 3x2−5x+2 | ||
=∫ | =...rozkład na ułamki proste | |
| (x−2)(x2+3) |
| 3x2−5x+2 | A | Bx+C | A*(x2+3)+(x−2)(Bx+C) | ||||
= | + | = | |||||
| (x−2)(x2+3) | x−2 | x2+3 | (x−2)(x2+3) |
| 3x2−5x+2 | A*(x2+3)+(x−2)(Bx+C) | ||
= | = | ||
| (x−2)(x2+3) | (x−2)(x2+3) |
| Ax2+3A+Bx2+Cx−2Bx−2C | ||
= | =grupujemy wyrazy i porównujemy liczniki | |
| (x−2)(x2+3) |
| f'(x | ||
∫ | dx=ln|f(x)|+C | |
| f(x) |
| 3 | 17 | 6 | ||||
A dobrze policzyłem współczynniki A,B,C ? A = 5 | B = − | C= −9 | ||||
| 7 | 7 | 7 |
http://www.wolframalpha.com/input/?i=Partial+fraction%28%283x%5E2+%E2%88%92+5x+%2B+2%29%2F%28x%5E3+%E2%88%92+2x%5E2+%2B+3x+%E2%88%92+6%29%29
| 4 | ||
A= | ||
| 7 |
| 17 | ||
B= | ||
| 7 |
| −1 | ||
C= | ||
| 7 |
| 4 | 1 |
| ||||||||||||||||
całka= | ∫ | dx+∫ | dx= | |||||||||||||||
| 7 | x−2 | x2+3 |
| 4 | 17 | 1 | 2x | 1 | 1 | |||||||
= | ln|x−2|+ | * | ∫ | dx− | ∫ | dx= | ||||||
| 7 | 7 | 2 | x2+3 | 7 | x2+3 |
| 4 | 17 | 1 | 1 | |||||
= | ln|x−2|+ | ln(x2+3)− | ∫ | dx= | ||||
| 7 | 14 | 7 | x2+3 |
| x | ||
[ostatnią całkę: x=√3t, t= | , dx=√3dt] | |
| √3 |
| 4 | 17 | √3 | x | |||||
= | ln|x−2|+ | ln(x2+3)− | arctg | +C | ||||
| 7 | 14 | 21 | √3 |