Laʻana o nā nīnau kūkākūkā hoʻohui palena ʻole
ʻO ka integral indefinite kahi manaʻo kumu i ka calculus, i hoʻohana ʻia e ʻimi i ka hana mua mai kahi hana i loaʻa. Hōʻike ʻia kahi integral indefinite e ka hōʻailona ∫ i ukali ʻia e ka hana e hoʻohui ʻia a me ka loli o ka hoʻohui ʻana. Ma kēia ʻatikala, e kūkākūkā mākou i kekahi mau laʻana o nā integrals indefinite a me kā lākou mau hoʻonā.
Laʻana Nīnau 1: Hoʻohui o nā Hana Polynomial
Nīnau: E hoʻoholo i ka hoʻohui o ka hana \( f(x) = 3x^2 \).
Kūkākūkā: No ka hoʻohui ʻana i nā hana polynomial, hoʻohana mākou i nā lula kumu o ka hoʻohui ʻana, ʻo ia hoʻi:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Ma ka hoʻohana ʻana i kēia mau lula, ʻo ka integral o \( 3x^2 \) penei:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) + C = 3 \left( \frac{1}{3} x^3 \right) + C = x^3 + C \]
No laila, \( \int 3x^2 \, dx = x^3 + C \).
Laʻana Nīnau 2: Hoʻohui o nā Hana Exponential
Nīnau: E hoʻoholo i ka hoʻohui o ka hana \( f(x) = e^x \).
Kūkākūkā: He maʻalahi loa ka hoʻohui o ka hana exponential \( e^x \) no ka mea, ʻo ka hana \( e^x \) he hana ia e kūpaʻa loa ana ma lalo o nā hana ʻokoʻa a me nā hana hoʻohui:
\[ \int e^x \, dx = e^x + C \]
No laila, \( \int e^x \, dx = e^x + C \).
Laʻana Nīnau 3: Hoʻohui o nā Hana Trigonometric
Nīnau: E hoʻoholo i ka integral o ka hana \( f(x) = \sin(x) \).
Kūkākūkā: No ka hoʻohui ʻana i nā hana trigonometric, pono mākou e ʻike i nā integrals kumu o kēlā mau hana. ʻO kekahi o nā pilina kumu:
\[ \int \sin(x) \, dx = -\cos(x) + C \]
No laila, \( \int \sin(x) \, dx = -\cos(x) + C \).
Laʻana Nīnau 4: Hoʻohui o nā Hana Hapa
Nīnau: E hoʻoholo i ka hoʻohui o ka hana \( f(x) = \frac{1}{x} \).
Kūkākūkā: ʻO ka hoʻohui o ka hana \( \frac{1}{x} \) penei:
\[ \int \frac{1}{x} \, dx = \ln|x| +C\]
No laila, \( \int \frac{1}{x} \, dx = \ln|x| + C \).
Laʻana Nīnau 5: Hoʻohui o nā Hana Exponential Maikaʻi ʻole
Nīnau: E hoʻoholo i ka hoʻohui o ka hana \( f(x) = x^{-2} \).
Kūkākūkā: No \( n \neq -1 \), hoʻohana mākou i ke kānāwai hoʻohui kumu:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Ma kēia hihia, \( n = -2 \), no laila:
\[ \int x^{-2} \, dx = \int x^{-2} \, dx = \frac{1}{-2+1} x^{-2+1} + C = \frac{1}{-1} x^{-1} + C = -x^{-1} + C = -\frac{1}{x} + C \]
No laila, \( \int x^{-2} \, dx = -\frac{1}{x} + C \).
Laʻana Nīnau 6: Hoʻohui o nā Hana Hoʻohuihui
Nīnau: E hoʻoholo i ka hoʻohui o ka hana \( f(x) = 4x^3 – 3x^2 + 2x – 5 \).
Kūkākūkā: Hiki iā mākou ke hoʻohui i kēlā me kēia huaʻōlelo ma ke kaʻawale me ka hoʻohana ʻana i nā lula kumu o ka hoʻohui ʻana:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = \int 4x^3 \, dx – \int 3x^2 \, dx + \int 2x \, dx – \int 5 \, dx \]
I kēia manawa, hoʻohui mākou i kēlā me kēia huaʻōlelo pākahi:
\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{1}{3+1} x^{3+1} \right) = 4 \left( \frac{1}{4} x^4 \right) = x^4 \]
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) = 3 \left( \frac{1}{3} x^3 \right) = x^3 \]
\[ \int 2x \, dx = 2 \int x \, dx = 2 \left( \frac{1}{1+1} x^{1+1} \right) = 2 \left( \frac{1}{2} x^2 \right) = x^2 \]
\[ \int 5 \, dx = 5x \]
Ke hoʻohui pū nei i kēia mau hopena, loaʻa iā mākou:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \]
No laila, \( \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \).
Ka hopena
He manaʻo koʻikoʻi loa ka integral palena ʻole i ka calculus a he mau lula like ʻole e maʻalahi ai ka hoʻohui ʻana i nā ʻano hana like ʻole. Ma kēia ʻatikala, ua kūkākūkā mākou i kekahi mau laʻana o nā integrals palena ʻole, me nā polynomials, exponents, nā hana trigonometric, nā hakina, nā hana me nā exponents maikaʻi ʻole, a me nā hui pū ʻana o nā hana. ʻO ka hoʻomaopopo ʻana a me ka hoʻopaʻa ʻana i kēia mau lula kumu o nā integrals e kōkua nui i ka hoʻoponopono ʻana i nā pilikia calculus like ʻole.
ʻAʻole wale nā integral palena ʻole he mea nui i ke kumumanaʻo makemakika, akā he nui nā noi i ka physics, ka ʻenekinia, a me nā ʻoihana ʻē aʻe. Me ka lawa o ka hoʻomaʻamaʻa, e maʻalahi a ʻoi aku ka maʻalahi o ka hoʻohui ʻana i nā hana like ʻole.
