Nā nīnau hoʻohālike e kūkākūkā ana i nā waiwai o nā Integrals Indefinite
He manaʻo koʻikoʻi ka integral indefinite i ka calculus, kahi e pili ana i ke kaʻina hana o ka loaʻa ʻana o ka hana mua mai kahi derivative i hāʻawi ʻia. Ua kapa pinepine ʻia kēia kaʻina hana he antiderivative a i ʻole ka hoʻohui ʻana. ʻO kahi hiʻohiʻona kū hoʻokahi o ka integral indefinite ʻo ia ka hopena o ka hoʻohui ʻana e hoʻokomo mau i kahi kūpaʻa o ka hoʻohui ʻana \(C \) no ka mea ʻo ka ʻokoʻa o kahi kūpaʻa he zero. E kūkākūkā kēia ʻatikala i kekahi mau laʻana o nā integrals indefinite a kūkākūkā i nā waiwai e pili ana me lākou.
1. Wehewehena o ka Indefinite Integral
ʻO ka hoʻohui palena ʻole o kahi hana \( f(x) \) he hana \( F(x) \) nona ka derivative e like me \( f(x) \). Ma ke ʻano hōʻailona, inā \( F'(x) = f(x) \), a laila:
\[
f(x) \, dx = F(x) + C
\]
kahi ʻo \( C \) ke kūpaʻa o ka hoʻohui ʻana.
2. Nā Waiwai o nā Hoʻohui Pau ʻole
No ka hoʻomaʻalahi ʻana i ke kaʻina hana hoʻohui, hiki iā mākou ke hoʻohana i kekahi mau waiwai laulā o nā integrals indefinite:
1. Nā Waiwai Linearity:
\[
\int [af(x) + bg(x)] \, dx = a \int f(x) \, dx + b \int g(x) \, dx
\]
kahi ʻo \( a \) a me \( b \) he mau mea mau.
2. Hoʻohui o ke Kūpaʻa:
\[
\int k \, dx = kx + C
\]
kahi he mau paʻa ʻo \( k \).
3. Hoʻohui o nā Mana:
\[
ʻO x^n, dx = x^{n+1}}{n+1} + C
\]
no \( n \neq -1 \).
4. Hoʻokaʻawale Hoʻohui:
\[
(f(x) + g(x)) \, dx = \, f(x) \, dx + \, g(x) \, dx
\]
Ma ka hoʻohana ʻana i kēia mau waiwai, hiki iā mākou ke hoʻoponopono i nā ʻano pilikia hoʻohui palena ʻole like ʻole.
3. Nā nīnau hoʻohālike a me ke kūkākūkā ʻana
Laʻana Nīnau 1: Hoʻohui o kahi hana quadratic
Nīnau: E hoʻoholo i ka integral o \( f(x) = 3x^2 \).
Kūkākūkā:
Hoʻohana mākou i ka waiwai hoʻohui o nā mana.
\[
3x^2 \, dx
\]
\[
= 3 \int x^2 \, dx
\]
Ma ka hoʻohana ʻana i nā waiwai integral:
\[
ʻO x^2, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}
\]
No laila:
\[
3 \int x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3
\]
Mai poina e hoʻohui i ke kūpaʻa hoʻohui:
\[
3x^2, dx = x^3 + C
\]
Laʻana Nīnau 2: Nā hoʻohui o nā hana trigonometric
Nīnau: E hoʻoholo i ka integral o \( f(x) = \sin(x) \).
Kūkākūkā:
Hoʻohana mākou i ka waiwai ʻo ka integral o \( \sin(x) \) ʻo \( -\cos(x) \):
\[
ʻO ka huaʻōlelo "sin(x)", dx = -cos(x) + C
\]
No laila:
\[
ʻO ka huaʻōlelo "sin(x)", dx = -cos(x) + C
\]
Laʻana 3: Hoʻohui o kahi hana exponential
Nīnau: E hoʻoholo i ka integral o \( f(x) = e^x \).
Kūkākūkā:
ʻO ka integral o \( e^x \) ʻo ia nō \( e^x \) no ka mea ua like nā ʻano o nā derivatives a me nā integral exponential:
\[
ʻO e^x, dx = e^x + C
\]
Laʻana Nīnau 4: Hoʻohui o kahi hana i hui pū ʻia
Nīnau: E hoʻoholo i ka integral o \( f(x) = x^2 + 3x + 1 \).
Kūkākūkā:
Hiki iā mākou ke hoʻohana i nā waiwai o ka hoʻokaʻawale integral:
\[
\int (x^2 + 3x + 1) \, dx = \int x^2 \, dx + \int 3x \, dx + \int 1 \, dx
\]
Ke hoʻohana nei i nā waiwai hoʻohui o kēlā me kēia ʻāpana:
\[
\int x^2 \, dx = \frac{x^3}{3}
\]
\[
\int 3x \, dx = 3 \int x \, dx = 3 \cdot \frac{x^2}{2} = \frac{3x^2}{2}
\]
\[
1, dx = x
\]
No laila:
\[
\int (x^2 + 3x + 1) \, dx = \frac{x^3}{3} + \frac{3x^2}{2} + x + C
\]
Laʻana Nīnau 5: Hoʻohui me ka pani maʻalahi
Nīnau: E hoʻoholo i ka integral o \( f(x) = (2x + 3)^5 \).
Kūkākūkā:
Maanei hiki ke hoʻohana ʻia ka pani \( u = 2x + 3 \). E huli i ka derivative \( du \):
\[
du = 2 \, dx \implies dx = \frac{1}{2} \, du
\]
No laila, lilo ka integral i:
\[
\int (2x + 3)^5 \, dx = \int u^5 \cdot \frac{dx}{du} \, du = \int u^5 \cdot \frac{1}{2} \, du = \frac{1}{2} \int u^5 \, du
\]
Hoʻohui ʻana \( u^5 \):
\[
\int u^5 \, du = \frac{u^6}{6}
\]
No laila, ʻo ka hopena hope loa:
\[
\frac{1}{2} \cdot \frac{u^6}{6} = \frac{u^6}{12}
\]
Ke pani nei iā \( u \) me \( 2x + 3 \):
\[
\frac{(2x + 3)^6}{12} + C
\]
Laʻana Nīnau 6: Hoʻohui o kahi hana hapa
Nīnau: E hoʻoholo i ka integral o \( f(x) = \frac{1}{x} \).
Kūkākūkā:
Ua ʻike mākou ʻo ka hoʻohui o \( \frac{1}{x} \) ʻo \( \ln{|x|} \):
\[
\int \frac{1}{x} \, dx = \ln{|x|} + C
\]
4. Manaʻo
He mea hana koʻikoʻi loa ka integral indefinite i ka calculus no ka loaʻa ʻana o ka hana mua mai kahi derivative i ʻike ʻia. He mea kōkua nui nā waiwai o ka linearity, ka integral o kahi constant, ka distributivity o nā integrals, a me nā mea ʻē aʻe i ke kaʻina hana hoʻohui. Me ka lawa ʻana o ka hoʻomaʻamaʻa, hiki ke hoʻoponopono pono ʻia nā ʻano integrals like ʻole.
Ma ka hoʻomaopopo ʻana i nā manaʻo kumu a me nā waiwai o nā integrals palena ʻole, ua manaʻolana ʻia e maʻalahi i nā haumāna ke hoʻoponopono i nā pilikia like ʻole e pili ana i nā integrals palena ʻole. ʻO ka hoʻomaʻamaʻa mau ʻana e hoʻoikaika i ko lākou ʻike a me ka hiki ke hoʻohana i nā integrals palena ʻole i nā ʻano makemakika like ʻole.
