Mohlala oa Lipotso tsa Puisano e Kopaneng e sa Feleng
Karolo e sa lekanyetsoang ke mohopolo oa motheo ka har'a calculus, o sebelisetsoang ho fumana mosebetsi oa pele ho tsoa mosebetsing o nkiloeng. Karolo e sa lekanyetsoang e bontšoa ke letšoao ∫ le lateloang ke mosebetsi o tla kopanngoa le phetoho ea kopanyo. Sehloohong sena, re tla tšohla mehlala e 'maloa ea likarolo tse sa lekanyetsoang le litharollo tsa tsona.
Mohlala oa Potso ea 1: Kakaretso ea Mesebetsi ea Polynomial
Potso: Fumana karolo e ka sehloohong ea mosebetsi \( f(x) = 3x^2 \).
Puisano: Ho kopanya mesebetsi ea polynomial, re sebelisa melao ea motheo ea kopanyo, e leng:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Ha re sebelisa melao ena, karolo ea bohlokoa ea \( 3x^2 \) ke:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) + C = 3 \left( \frac{1}{3} x^3 \right) + C = x^3 + C \]
Kahoo, \( \int 3x^2 \, dx = x^3 + C \).
Mohlala oa Potso ea 2: Kakaretso ea Mesebetsi ea Exponential
Potso: Fumana karolo e ka sehloohong ea mosebetsi \( f(x) = e^x \).
Puisano: Karolo e kopaneng ea mosebetsi oa exponential \( e^x \) e bonolo haholo hobane mosebetsi \( e^x \) ke mosebetsi o sa fetoheng ka ho feletseng tlas'a mesebetsi e fapaneng le e kopaneng:
\[ \int e^x \, dx = e^x + C \]
Kahoo, \( \int e^x \, dx = e^x + C \).
Mohlala oa Potso ea 3: Mesebetsi e Kopanetsoeng ea Trigonometric
Potso: Fumana karolo e ka sehloohong ea mosebetsi \( f(x) = \sin(x) \).
Puisano: Ho kopanya mesebetsi ea trigonometric, re hloka ho tseba likarolo tsa motheo tsa mesebetsi eo. E 'ngoe ea likamano tsa motheo ke:
\[ \int \sin(x) \, dx = -\cos(x) + C \]
Kahoo, \( \int \sin(x) \, dx = -\cos(x) + C \).
Mohlala oa Potso ea 4: Mesebetsi e Kopanetsoeng ea Likaroloana
Potso: Fumana karolo e feletseng ea mosebetsi \( f(x) = \frac{1}{x} \).
Puisano: Karolo ya mosebetsi \( \frac{1}{x} \) ke:
\[ \int \frac{1}{x} \, dx = \ln|x| +C\]
Kahoo, \( \int \frac{1}{x} \, dx = \ln|x| + C \).
Mohlala oa Potso ea 5: Kakaretso ea Mesebetsi e Mebe ea Tlhaloso
Potso: Fumana karolo e ka sehloohong ea mosebetsi \( f(x) = x^{-2} \).
Puisano: Bakeng sa \( n \neq -1 \), re sebedisa molao wa motheo wa integral:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Tabeng ena, \( n = -2 \), kahoo:
\[ \int x^{-2} \, dx = \int x^{-2} \, dx = \frac{1}{-2+1} x^{-2+1} + C = \frac{1}{-1} x^{-1} + C = -x^{-1} + C = -\frac{1}{x} + C \]
Kahoo, \( \int x^{-2} \, dx = -\frac{1}{x} + C \).
Mohlala oa Potso ea 6: Mesebetsi e Kopantsoeng e Kopantsoeng
Potso: Fumana karolo e ka sehloohong ea mosebetsi \( f(x) = 4x^3 – 3x^2 + 2x – 5 \).
Puisano: Re ka kopanya poleloana ka 'ngoe ka thoko re sebelisa melao ea motheo ea kopanyo:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = \int 4x^3 \, dx – \int 3x^2 \, dx + \int 2x \, dx – \int 5 \, dx \]
Jwale re kopanya polelo ka nngwe ka bonngoe:
\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{1}{3+1} x^{3+1} \right) = 4 \left( \frac{1}{4} x^4 \right) = x^4 \]
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) = 3 \left( \frac{1}{3} x^3 \right) = x^3 \]
\[ \int 2x \, dx = 2 \int x \, dx = 2 \left( \frac{1}{1+1} x^{1+1} \right) = 2 \left( \frac{1}{2} x^2 \right) = x^2 \]
\[ \int 5 \, dx = 5x \]
Ha re kopanya liphetho tsena, re fumana:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \]
Kahoo, \( \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \).
Qetello
Karolo e sa lekanyetsoang ke mohopolo oa bohlokoa haholo ho calculus 'me e na le melao e fapaneng e etsang hore ho be bonolo ho kopanya mefuta e fapaneng ea mesebetsi. Sehloohong sena, re buisane ka mehlala e 'maloa ea likarolo tse sa lekanyetsoang, ho kenyeletsoa le li-polynomial, li-exponential, mesebetsi ea trigonometric, likaroloana, mesebetsi e nang le li-exponents tse mpe, le motsoako oa mesebetsi. Ho utloisisa le ho tseba melao ena ea motheo ea likarolo tse kopaneng ho tla thusa haholo ho rarolla mathata a fapaneng a calculus.
Di-integral tse sa feleng ha di bohlokwa feela thutong ya dipalo, empa hape di na le ditshebediso tse pharaletseng fisiks, boenjiniere le mafapheng a mang. Ka ho ikwetlisa ho lekaneng, ho kopanya mesebetsi e fapaneng ho tla ba bonolo le ho ba bonolo haholoanyane.
