Kalkulator całek
Ten kalkulator całek (dla wielomianów) dokładnie znajduje funkcje pierwotne i całki oznaczone, pokazując wzór na całkę potęgi dla każdego wyrazu.
Wynik
Wpisz wartości i naciśnij Oblicz, aby zobaczyć wynik.
Wzór
The reverse power rule (for n ≠ −1).
The fundamental theorem of calculus.
Jak korzystać z tego kalkulatora
- Type the polynomial and choose indefinite or definite.
- For a definite integral, enter the lower and upper limits.
Rozwiązane przykłady
∫ from 0 to 2 of 3x² dx
An antiderivative of 3x² is x³.
F(2) − F(0) = 8 − 0 = 8.
Easy: an indefinite integral
Add 1 to each power and divide by the new power: x² → x³/3, and 4x → 4x²/2 = 2x².
∫(x² + 4x) dx = x³/3 + 2x² + C.
Hard: a definite integral with a negative limit
Antiderivative: F(x) = x³/3 − x² + 3x.
F(2) = 8/3 − 4 + 6 = 14/3 and F(−1) = −1/3 − 1 − 3 = −13/3.
∫ from −1 to 2 = F(2) − F(−1) = 14/3 + 13/3 = 27/3 = 9.
Częste błędy
- Forgetting + C on an indefinite integral.
- Subtracting the wrong way: a definite integral is F(upper) − F(lower).
- Sign slips at a negative limit: F(−1) = −1/3 − 1 − 3, and subtracting it adds 13/3.
Najczęstsze pytania
Can it do partial fraction integration?
Not directly — it integrates polynomials. For a fraction such as 1/((x − 1)(x + 2)), split it first with the Partial Fractions Calculator; each simple fraction then integrates to a logarithm.
What is + C?
Any constant differentiates to zero, so every antiderivative is defined up to a constant C.
Can it integrate sin x or eˣ?
This calculator handles polynomials. Sumzo AI can help with other functions.