Ihe atụ nke Ajụjụ Mkparịta ụka Integral
Integral bụ echiche dị mkpa na mgbakọ na mwepụ nke nwere ọtụtụ ọrụ n'ọtụtụ ngalaba, gụnyere physics, injinia, na akụnụba. Isiokwu a ga-enyocha ọtụtụ ihe atụ nke nsogbu dị mkpa na ngwọta ha iji nye nghọta miri emi.
1. Nghọta Isi nke Integrals
N'ikwu ya n'ụzọ dị mfe, integral bụ ọrụ ntụgharị nke ihe mepụtara. E nwere ụdị integral abụọ a na-ekwukarị, ya bụ:
– Integral Na-adịghị Agbanwe Agbanwe: nke a bụ ụdị dị mkpa nke na-enweghị oke elu na ala ma a na-egosi ya site na ∫ f(x) dx.
– Nkọwapụta nke zuru oke: nke a bụ ụdị dị mkpa nke nwere oke elu na ala ma ∫[a,b] f(x) dx na-egosi ya.
A na-akpọkarị ihe na-adịghị agbanwe agbanwe nke na-emegide ihe na-akpata nsogbu (anti-derivative), ihe ga-esi na ya pụta ga-agụnye ihe na-agbanwe agbanwe nke C n'ihi na ihe onwunwe nke ihe na-agbanwe agbanwe bụ efu.
2. Ihe atụ nke Nsogbu Ndị A Na-apụghị Ịhụ Anya
Ihe atụ nke 1: Mfe Indefinite Integral
Gbakọọ ∫ x^2 dx.
Azịza:
Anyị maara na iwu njikọta bụ isi maka ∫ x^n dx bụ (x^(n+1))/(n+1) + C, ebe C bụ ihe na-agbanwe agbanwe nke njikọta.
Maka ihe mejupụtara dị n'elu, n = 2:
∫ x^2 dx = (x^(2+1))/(2+1) + C
= (x^3)/3 + C.
Ya mere, ihe si na ∫ x^2 dx pụta bụ (x^3)/3 + C.
Ihe atụ nke abụọ: Njikọta nke Ọrụ Exponential
Gbakọọ ∫ e^x dx.
Azịza:
Iwu bụ isi maka ihe mgbakwunye ∫ e^x dx bụ e^x + C.
Ya mere, ihe si na ∫ e^x dx pụta bụ e^x + C.
3. Ihe atụ nke Nsogbu Ndị A Na-ahụ Anya
Ihe atụ nke 1: Mfe Nkọwapụta Dị Mfe
Gbakọọ ∫[1,3] x^2 dx.
Azịza:
Nke mbụ, anyị na-ahụ ihe mgbochi nke x^2, nke bụ (x^3)/3.
Ugbu a, anyị na-agbanwe ihe mgbochi ndị a:
∫[1,3] x^2 dx = [(3^3)/3 – (1^3)/3]
= [27/3 – 1/3]
= [9 – 1/3]
= 8 + 2/3 ma ọ bụ 8.6667.
Ya mere, ihe si na ∫[1,3] x^2 dx pụta bụ 26/3 ma ọ bụ 8.6667.
Ihe atụ nke 2: Integral site na Mgbanwe
Gbakọọ ∫[0,2] (2x + 1) dx.
Azịza:
Nke mbụ, anyị na-achọta ihe mgbochi nke 2x + 1, nke bụ x^2 + x. Ugbu a, anyị na-edochi ihe mgbochi ndị a:
∫[0,2] (2x+1) dx = [(2^2 + 2) – (0^2 + 0)]
= [(4 + 2) – 0]
= 6.
Ya mere, nsonaazụ nke ∫[0,2] (2x + 1) dx bụ 6.
4. Ihe atụ nke Nsogbu Ndị Dị n'Otu na Usoro Nkịtị
Usoro nkewa nke ihe mejupụtara akụkụ bụ usoro eji eme ihe mgbe ọ siri ike ịgbakọ ihe mejupụtara ngwaahịa nke ọrụ abụọ ozugbo. Usoro maka ihe mejupụtara akụkụ bụ:
∫ u dv = uv – ∫ v du
Ihe atụ: Ihe mejupụtara akụkụ nke Trigonometric
Gbakọọ ∫ xe^x dx.
Azịza:
N'ebe a, anyị na-eji usoro nkebi. Ka e were ya na u = x na dv = e^x dx. Mgbe ahụ du = dx na v = e^x.
Dabere na usoro nkewa nke akụkụ:
∫ xe^x dx = xe^x – ∫ e^x dx
= xe^x – e^x + C
= e^x(x – 1) + C.
Ya mere, ihe si na ∫ xe^x dx pụta bụ e^x(x – 1) + C.
5. Ihe atụ nke nsogbu Trigonometric Integral
Ihe atụ: Integral nke Ọrụ Trigonometric Isi
Gbakọọ ∫ cos(x) dx.
Azịza:
Iwu bụ isi nke njikọta nke cos(x) bụ sin(x) + C.
Ya mere, ihe si na ∫ cos(x) dx pụta bụ sin(x) + C.
Ihe atụ: Integral nke ọrụ Trigonometric nwere oke
Gbakọọ ∫[0,π/2] sin(x) dx.
Azịza:
Nke mbụ, anyị na-ahụ ihe na-emegide sin(x), nke bụ -cos(x).
Ugbu a, dochie ihe mgbochi ndị a:
∫[0,π/2] mmehie (x) dx = [-cos (π/2) - (-cos (0))]
= [ -0 – (-1) ]
= 1.
Ya mere, ihe si na ∫[0,π/2] sin(x) dx pụta bụ 1.
6. Ihe atụ nke Nsogbu Njikọta Ndochi
Ihe atụ: Nchikota ihe nnọchi anya
Gbakọọ ∫ 2x sqrt(1-x^2) dx.
Azịza:
Jiri mgbanwe u = 1-x^2, wee jiri du = -2x dx.
Mgbe ahụ, ihe dị mkpa ga-agbanwe gaa na:
∫ sqrt(u) (-1/2 du)
= -1/2 ∫ u^(1/2) du
= -1/2 [ (2/3) u^(3/2) ] + C
= -1/3 (1-x^2)^(3/2) + C.
Ya mere, ihe si na ∫ 2x sqrt(1-x^2) dx pụta bụ -1/3 (1-x^2)^(3/2) + C.
7. Kesimpulun
Ngwakọta bụ ngwa bara uru nke ukwuu na mgbakọ na mwepụ maka ịchọta mpaghara dị n'okpuru mgbagọ, olu, na ọtụtụ ngwa ndị ọzọ. Ịghọta usoro njikọta dị iche iche, dị ka nnọchi, akụkụ, na ntọala nke njikọta, dị oke mkpa. Ihe atụ ndị a tụlere n'elu ga-enyere gị aka ịghọta echiche nke njikọta.
Omume na nghọta echiche mgbe niile dị mkpa iji bụrụ ọkachamara n'ihe gbasara njikọta. Gaa n'ihu na-eme ihe na mgbanwe dị iche iche na ụdị ọrụ dị iche iche iji gbasaa ihe ọmụma gị n'akụkụ a.
Enwere m olileanya na isiokwu a ga-abara gị uru n'ịmụta ihe dị iche iche.
