Chitsanzo cha Mafunso Okambirana Osatha
Chigwirizano chosatha ndi lingaliro lofunikira mu calculus, lomwe limagwiritsidwa ntchito kupeza ntchito yoyambirira kuchokera ku ntchito yochokera. Chigwirizano chosatha chimawonetsedwa ndi chizindikiro ∫ kutsatiridwa ndi ntchito yoti iphatikizidwe ndi chosinthika cha kuphatikizana. M'nkhaniyi, tikambirana zitsanzo zingapo za zigwirizano zosatha ndi mayankho awo.
Chitsanzo Funso 1: Kuphatikiza kwa Ntchito za Polynomial
Funso: Dziwani kufunika kwa ntchito \( f(x) = 3x^2 \).
Kukambirana: Kuti tiphatikize ntchito za polynomial, timagwiritsa ntchito malamulo oyambira ophatikizana, omwe ndi:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Pogwiritsa ntchito malamulo awa, mfundo yofunika kwambiri ya \( 3x^2 \) ndi:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) + C = 3 \left( \frac{1}{3} x^3 \right) + C = x^3 + C \]
Kotero, \( \int 3x^2 \, dx = x^3 + C \).
Chitsanzo Funso 2: Kuphatikiza kwa Ntchito Zowonetsera
Funso: Dziwani kufunika kwa ntchito \( f(x) = e^x \).
Kukambirana: Kuphatikizika kwa ntchito ya exponential \( e^x \) n'kosavuta chifukwa ntchito \( e^x \) ndi ntchito yomwe siisintha konse pansi pa ntchito zosiyana komanso zophatikizika:
\[ \int e^x \, dx = e^x + C \]
Kotero, \( \int e^x \, dx = e^x + C \).
Chitsanzo Funso 3: Kuphatikiza Ntchito za Trigonometric
Funso: Dziwani kufunika kwa ntchito \( f(x) = \sin(x) \).
Kukambirana: Kuti tigwirizane ndi ntchito za trigonometric, tifunika kudziwa zinthu zofunika kwambiri za ntchito zimenezo. Limodzi mwa maubwenzi oyambira ndi:
\[ \int \sin(x) \, dx = -\cos(x) + C \]
Kotero, \( \int \sin(x) \, dx = -\cos(x) + C \).
Chitsanzo Funso 4: Kuphatikiza kwa Ntchito Zogawika
Funso: Dziwani integral ya ntchito \( f(x) = \frac{1}{x} \).
Kukambirana: Chigawo chofunikira cha ntchito \( \frac{1}{x} \) ndi:
\[ \int \frac{1}{x} \, dx = \ln|x| +C\]
Kotero, \( \int \frac{1}{x} \, dx = \ln|x| + C \).
Chitsanzo Funso 5: Kuphatikiza kwa Ntchito Zoyipa Zowonetsera
Funso: Dziwani integral ya ntchito \( f(x) = x^{-2} \).
Kukambirana: Pa \( n \neq -1 \), timagwiritsa ntchito lamulo lofunikira la integral:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Pankhaniyi, \( n = -2 \), kotero:
\[ \int x^{-2} \, dx = \int x^{-2} \, dx = \frac{1}{-2+1} x^{-2+1} + C = \frac{1}{-1} x^{-1} + C = -x^{-1} + C = -\frac{1}{x} + C \]
Kotero, \( \int x^{-2} \, dx = -\frac{1}{x} + C \).
Chitsanzo Funso 6: Kuphatikiza Ntchito Zophatikizana
Funso: Dziwani integral ya ntchito \( f(x) = 4x^3 – 3x^2 + 2x – 5 \).
Kukambirana: Tikhoza kuphatikiza liwu lililonse padera pogwiritsa ntchito malamulo oyambira ophatikiza:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = \int 4x^3 \, dx – \int 3x^2 \, dx + \int 2x \, dx – \int 5 \, dx \]
Tsopano tikuphatikiza liwu lililonse payekhapayekha:
\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{1}{3+1} x^{3+1} \right) = 4 \left( \frac{1}{4} x^4 \right) = x^4 \]
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) = 3 \left( \frac{1}{3} x^3 \right) = x^3 \]
\[ \int 2x \, dx = 2 \int x \, dx = 2 \left( \frac{1}{1+1} x^{1+1} \right) = 2 \left( \frac{1}{2} x^2 \right) = x^2 \]
\[ \int 5 \, dx = 5x \]
Kuphatikiza zotsatirazi, timapeza:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \]
Kotero, \( \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \).
Mapeto
Chigwirizano chosatha ndi lingaliro lofunika kwambiri mu calculus ndipo chili ndi malamulo osiyanasiyana omwe amapangitsa kuti zikhale zosavuta kuphatikiza mitundu yosiyanasiyana ya ntchito. M'nkhaniyi, takambirana zitsanzo zingapo za zigwirizano zosatha, kuphatikizapo ma polynomial, ma exponential, ntchito za trigonometric, magawo, ntchito zokhala ndi ma exponents olakwika, ndi kuphatikiza ntchito. Kumvetsetsa ndikudziwa bwino malamulo oyambira awa a zigwirizano kudzakuthandizani kwambiri kuthetsa mavuto osiyanasiyana a calculus.
Ma integral osatha si ofunikira kokha mu chiphunzitso cha masamu, komanso amagwiritsidwa ntchito kwambiri mu fizikisi, uinjiniya, ndi madera ena. Ndi machitidwe okwanira, kuphatikiza ntchito zosiyanasiyana kudzakhala kosavuta komanso kosavuta.
