Ntinye Trigonometric Integral
Integrals bụ isi echiche na calculus, nke a na-eji agbakọ mpaghara, olu nke ihe siri ike nke mgbanwe, ogologo mgbagọ, na nsogbu dị iche iche nke physics dịka ọrụ na ike. N'omume, ọ bụghị ihe niile nwere ike idozi ozugbo. E nwere ụfọdụ ihe ndị yiri "ọgwụgwụ" ma ọ bụrụ na ejiri naanị iwu ndị bụ isi. Nke a bụ ebe usoro nnọchi anya na-aghọ ihe dị mkpa. Otu usoro nnọchi anya dị ike ma na-ejikarị eme ihe bụ mgbanwe trigonometric, usoro nke ịtụgharị okwu algebra (karịsịa ndị metụtara radicals) ka ọ bụrụ ụdị trigonometric iji mee ka ihe dị mfe.
1. Gịnị bụ mgbanwe trigonometric?
Ndozigharị nke trigonometric bụ usoro njikọta nke na-eji njirimara trigonometric iji mee ka ihe ndị nwere radicals dị mfe dịka:
– \(\sqrt{a^2 – x^2}\)
– \(\sqrt{a^2 + x^2}\)
– \(\sqrt{x^2 – a^2}\)
Isi echiche: anyị na-eji okwu nke gụnyere ọrụ trigonometric (dịka ọmụmaatụ \(x = a\sin\theta\), \(x = a\tan\theta\), ma ọ bụ \(x = a\sec\theta\)) dochie \(x\) ka mgbọrọgwụ ndị dị mgbagwoju anya wee bụrụ ụdị nke dị mfe iji site na iji njirimara trigonometric.
Usoro a dị irè n'ihi na njirimara dịka:
– \(1 – \sin^2\theta = \cos^2\theta\)
- \ (1 + \ tan^2 \ theta = \ sec^2 \ theta \)
- (\ sec ^ 2 isiokwu - 1 = tan ^ 2 "theta \)
ime ka mgbọrọgwụ square ghọọ ọrụ trigonometric dị mfe, ọtụtụ mgbe naanị \(\cos\theta\), \(\sec\theta\), ma ọ bụ \(\tan\theta\).
2. Olee mgbe a na-eji mgbanwe trigonometric eme ihe?
Ndochi trigonometric na-akacha mma mgbe integral nwere mgbọrọgwụ sụkwịa nke ụdị quadratic. Ọ dị mkpa ịmata ụkpụrụ izugbe atọ:
1. Ụdị \(\sqrt{a^2 – x^2}\)
Dabara adaba maka iji nnọchi:
\[
x = a\sin\theta \quad \Rightarrow \quad \sqrt{a^2 – x^2} = a\cos\theta
\]
N'ihi na:
\[
a^2 – a^2\sin^2\theta = a^2(1-\sin^2\theta)=a^2\cos^2\theta
\]
2. Ụdị \(\sqrt{a^2 + x^2}\)
Dabara adaba maka iji nnọchi:
\[
x = a\tan\theta \quad \Rightarrow \quad \sqrt{a^2 + x^2} = a\sec\theta
\]
N'ihi na:
\[
a^2 + a^2\tan^2\theta = a^2(1+\tan^2\theta)=a^2\sec^2\theta
\]
3. Ụdị \(\sqrt{x^2 – a^2}\)
Dabara adaba maka iji nnọchi:
\[
x = a\sec\theta \quad \Rightarrow \quad \sqrt{x^2 – a^2} = a\tan\theta
\]
N'ihi na:
\[
a^2\sec^2\theta – a^2 = a^2(\sec^2\theta – 1)=a^2\tan^2\theta
\]
Site n'ịmata ụkpụrụ ndị a, anyị nwere ike ịhọrọ ozugbo nnọchi kwesịrị ekwesị na-enweghị nnwale na mmejọ dị ukwuu.
3. Usoro izugbe maka mkpebi
N'ozuzu, usoro mgbanwe trigonometric bụ:
1. Chọpụta ọdịdị mgbọrọgwụ nke dabara na otu n'ime ụkpụrụ ndị ahụ.
2. Chọpụta ihe nnọchi anya ya (dịka ọmụmaatụ \(x = a\sin\theta\)).
3. Gbakọọ ihe ndị e si na ya nweta: \(dx\) n'ụdị \(\theta\).
Ihe atụ: ọ bụrụ na \(x=a\sin\theta\), mgbe ahụ \(dx = a\cos\theta\,d\theta\).
4. Gbanwee ihe dị na integral ka ọ bụrụ integral dị na \(\theta\).
5. Dozie ihe dị na \(\theta\).
6. Laghachi na mgbanwe \(x\) site na iji triangle ma ọ bụ njirimara enyemaka dochie \(\theta\) ọzọ.
Nzọụkwụ ikpeazụ na-abụkarị ihe na-agbagwoju ụmụ akwụkwọ anya, mana enwere ike ime ka ọ dị mfe site n'iwulite triangle kwesịrị ekwesị n'etiti \(x\), \(a\), na mgbọrọgwụ sitere na ya.
4. Ihe atụ nke 1: Integral nke ụdị \(\sqrt{a^2-x^2}\)
Ihe Nlereanya:
\[
\int \sqrt{a^2-x^2}\,dx
\]
Jiri nnọchi anya:
\[
x = a\sin\theta,\quad dx = a\cos\theta\, d\theta
\]
Mgbe ahụ:
\[
\sqrt{a^2-x^2} = \sqrt{a^2-a^2\sin^2\theta}=a\cos\theta
\]
Mgbe ahụ, ihe dị mkpa ga-agbanwe gaa na:
\[
\int (a\cos\theta)(a\cos\theta)\,d\theta = a^2\int \cos^2\theta\,d\theta
\]
Jiri njirimara \(\cos^2\theta = \frac{1+\cos 2\theta}{2}\):
\[
a^2\int \frac{1+\cos 2\theta}{2}\,d\theta
= \frac{a^2}{2}\left(\theta+\frac{1}{2}\sin 2\theta\right)+C
\]
Mgbe nke ahụ gasịrị, laghachi na \(x\). Site na \(x=a\sin\theta\), anyị na-enweta:
\[
\theta = \arcsin\left(\frac{x}{a}\right)
\]
na \(\sin 2\theta = 2\sin\theta\cos\theta = 2\frac{x}{a}\frac{\sqrt{a^2-x^2}}{a}=\frac{2x\sqrt{a^2-x^2}}{a^2}\).
Nsonaazụ ikpeazụ:
\[
\int \sqrt{a^2-x^2}\,dx
= \frac{a^2}{2}\arcsin\left(\frac{x}{a}\right)+\frac{x}{2}\sqrt{a^2-x^2}+C
\]
Ụdị a na-apụtakarị na nsogbu na mpaghara nke gburugburu ma ọ bụ geometry.
5. Ihe atụ nke 2: Integral nke ụdị \(\sqrt{a^2+x^2}\)
Tụlee:
\[
\int \sqrt{x^2+a^2}\,dx
\]
Ndochi:
\[
x = a\tan\theta,\quad dx = a\sec^2\theta\,d\theta
\]
Ya mere:
\[
\sqrt{x^2+a^2}=\sqrt{a^2\tan^2\theta+a^2}=a\sec\theta
\]
Ihe dị n'ime ya na-aghọ:
\[
\int (a\sec\theta)(a\sec^2\theta)\,d\theta = a^2\int \sec^3\theta\,d\theta
\]
Usoro ihe mejupụtara \(\int \sec^3\theta d\theta\) nwere usoro iheomume ochie:
\[
\int \sec^3\theta\,d\theta = \frac{1}{2}\sec\theta\tan\theta+\frac{1}{2}\ln|\sec\theta+\tan\theta|+C
\]
Ka ọ were:
\[
\int \sqrt{x^2+a^2}\,dx = \frac{a^2}{2}\sec\theta\tan\theta+\frac{a^2}{2}\ln|\sec\theta+\tan\theta|+C
\]
Laghachi na \(x\). Ebe ọ bụ na \(x=a\tan\theta\), mgbe ahụ \(\tan\theta = x/a\) na \(\sec\theta=\sqrt{1+\tan^2\theta}=\sqrt{1+x^2/a^2}=\frac{\sqrt{x^2+a^2}}{a}\).
Nsonaazụ:
\[
\int \sqrt{x^2+a^2}\,dx
= \frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln\left|\frac{\sqrt{x^2+a^2}+x}{a}\right|+C
\]
Nke a na-apụtakarị na fiziki na ngụkọ ogologo nke mgbagọ.
6. Ndụmọdụ dị mkpa iji zere mmejọ
1. Họrọ nnọchi dịka ụdị radical si dị. Ejila \(x=a\sin\theta\) maka \(\sqrt{a^2+x^2}\), n'ihi na njirimara ahụ adabaghị.
2. Lezienụ anya na ngalaba ahụ. Dịka ọmụmaatụ, mgbe ị na-eji \(\theta=\arcsin(x/a)\), ọ na-abụkarị \(x\) ga-adị na \([-a,a]\) ka mgbọrọgwụ wee bụrụ eziokwu.
3. Jiri triangle enyemaka laghachi na \(x\).
Ihe atụ: ọ bụrụ na \(x=a\tan\theta\), wuo triangle aka nri nwere akụkụ nke ọzọ \(x\), akụkụ dị n'akụkụ \(a\), ka hypotenuse wee bụrụ \(\sqrt{x^2+a^2}\). Site n'ebe ahụ \(\sec\theta = \frac{\sqrt{x^2+a^2}}{a}\), \(\sin\theta=\frac{x}{\sqrt{x^2+a^2}}\), na ndị ọzọ.
4. Jiri nlezianya dochie \(dx\). Ọtụtụ njehie na-eme n'ihi ichefu ịgbanwe ọdịiche ahụ.
7. Mmechi
Ndozigharị Trigonometric bụ usoro bara uru nke ukwuu maka idozi ihe ndị mejupụtara ya gụnyere mgbọrọgwụ sụkwịa nke okwu quadratic. Site n'ịmata ụkpụrụ atọ bụ isi—\(\sqrt{a^2-x^2}\), \(\sqrt{a^2+x^2}\), na \(\sqrt{x^2-a^2}\)—anyị nwere ike ịhọrọ nnọchi kwesịrị ekwesị ma mee ka ihe mejupụtara ahụ dị mfe n'usoro. Ọ bụ ezie na usoro ndị a nwere ike iyi ogologo, omume na-aga n'ihu ga-eme ka usoro ahụ dịkwuo akpaka ma dị mfe. N'ikpeazụ, nnọchi trigonometric abụghị naanị "aghụghọ," kama ọ bụ atụmatụ mgbakọ na mwepụ nke na-eji ike nke njirimara trigonometric iji dozie nsogbu dị mgbagwoju anya.
Ọ bụrụ na ịchọrọ, enwere m ike itinye ngalaba pụrụ iche nke nwere ajụjụ omume (tinyere mkparịta ụka) ka isiokwu a dịkwa njikere iji dịka ihe mmụta.
