Karolo e Kopanetsoeng ea Trigonometric
Li-integral ke mohopolo oa mantlha ka har'a lipalo, o sebelisetsoang ho bala libaka, boholo ba lintho tse tiileng tsa phetoho, bolelele ba likhutlo, le mathata a fapaneng a fisiks joalo ka mosebetsi le matla. Ha e le hantle, ha se li-integral tsohle tse ka rarolloang ka kotloloho. Ho na le li-integral tse itseng tse bonahalang li "sa fella" haeba feela ho sebelisoa melao ea motheo ea integral. Mona ke moo mekhoa ea ho nkela sebaka e bang ea bohlokoa. Mokhoa o mong o matla le o sebelisoang khafetsa oa ho nkela sebaka ke trigonometric substitution, mokhoa oa ho fetola lipolelo tsa algebra (haholo-holo tse amanang le radicals) hore e be libopeho tsa trigonometric ho nolofatsa integral.
1. Phetolo ea trigonometric ke eng?
Phetolo ea Trigonometric ke mokhoa oa kopanyo o sebelisang boitsebiso ba trigonometric ho nolofatsa lintho tse kopaneng tse nang le li-radical tse kang:
– \(\sqrt{a^2 – x^2}\)
– \(\sqrt{a^2 + x^2}\)
– \(\sqrt{x^2 – a^2}\)
Khopolo-taba e ka sehloohong: re nkela sebaka sa \(x\) ka polelo e kenyeletsang mesebetsi ea trigonometric (mohlala, \(x = a\sin\theta\), \(x = a\tan\theta\), kapa \(x = a\sec\theta\)) e le hore metso e rarahaneng e fetohe sebopeho se bonolo ho se nolofatsa ka ho sebelisa boitsebiso ba trigonometric.
Mokhoa ona o sebetsa hantle hobane boitsebiso bo kang:
– \(1 – \sin^2\theta = \cos^2\theta\)
– \(1 + \tan^2\theta = \sec^2\theta\)
– \(\ sec^2\theta – 1 = \tan^2\theta\)
ho etsa hore motso o sekoere o fetohe mosebetsi o bonolo oa trigonometric, hangata e le feela \(\cos\theta\), \(\sec\theta\), kapa \(\tan\theta\).
2. Phetolo ea trigonometric e sebelisoa neng?
Phetolo ea Trigonometric e bonolo haholo ha motsoako o nang le motso o sekoere oa sebopeho sa quadratic. Ho bohlokoa ho lemoha mekhoa e meraro e akaretsang:
1. Sebopeho \(\sqrt{a^2 – x^2}\)
E loketse ho sebelisa lintho tse nkeloang sebaka:
\[
x = a\sin\theta \quad \Rightarrow \quad \sqrt{a^2 – x^2} = a\cos\theta
\]
Hobane:
\[
a^2 – a^2\sin^2\theta = a^2(1-\sin^2\theta)=a^2\cos^2\theta
\]
2. Sebopeho \(\sqrt{a^2 + x^2}\)
E loketse ho sebelisa lintho tse nkeloang sebaka:
\[
x = a\tan\theta \quad \Rightarrow \quad \sqrt{a^2 + x^2} = a\sec\theta
\]
Hobane:
\[
a^2 + a^2\tan^2\theta = a^2(1+\tan^2\theta)=a^2\sec^2\theta
\]
3. Sebopeho \(\sqrt{x^2 – a^2}\)
E loketse ho sebelisa lintho tse nkeloang sebaka:
\[
x = a\sec\theta \quad \Rightarrow \quad \sqrt{x^2 – a^2} = a\tan\theta
\]
Hobane:
\[
a^2\sec^2\theta – a^2 = a^2(\sec^2\theta – 1)=a^2\tan^2\theta
\]
Ka ho lemoha mekhoa ena, re ka khetha sebaka se loketseng hang-hang ntle le liteko le liphoso tse ngata.
3. Mehato e akaretsang ea tharollo
Ka kakaretso, mokhoa oa ho nkela sebaka oa trigonometric ke:
1. Hlalosa sebopeho sa motso se lumellanang le e 'ngoe ea lipaterone.
2. Fumana sebaka sa ho nkela sebaka (mohlala: \(x = a\sin\theta\)).
3. Bala derivative: \(dx\) ka sebopeho sa \(\theta\).
Mohlala: haeba \(x=a\sin\theta\), joale \(dx = a\cos\theta\,d\theta\).
4. Fetola karolo e kopaneng ho ba karolo e kopaneng ho \(\theta\).
5. Rarolla karolo e kopaneng ho \(\theta\).
6. Khutlela ho feto-fetoha \(x\) ka ho nkela \(\theta\) sebaka hape o sebedisa kgutlotharo e thusang kapa boitsebiso.
Mohato wa ho qetela hangata ke wona o ferekanyang baithuti ka ho fetisisa, empa o ka nolofatswa ka ho aha khutlotharo e loketseng pakeng tsa \(x\), \(a\), le motso o hlahang.
4. Mohlala oa 1: Karolo e kopaneng ea foromo \(\sqrt{a^2-x^2}\)
Mohlala:
\[
\int \sqrt{a^2-x^2}\,dx
\]
Sebelisa sebaka:
\[
x = a\sin\theta,\quad dx = a\cos\theta\, d\theta
\]
Ebe:
\[
\sqrt{a^2-x^2} = \sqrt{a^2-a^2\sin^2\theta}=a\cos\theta
\]
Ebe phetoho ea bohlokoa e fetoha ho:
\[
\int (a\cos\theta)(a\cos\theta)\,d\theta = a^2\int \cos^2\theta\,d\theta
\]
Sebelisa boitsebiso \(\cos^2\theta = \frac{1+\cos 2\theta}{2}\):
\[
a^2\int \frac{1+\cos 2\theta}{2}\,d\theta
= \frac{a^2}{2}\left(\theta+\frac{1}{2}\sin 2\theta\right)+C
\]
Ka mor'a moo, khutlela ho \(x\). Ho tloha ho \(x=a\sin\theta\), re fumana:
\[
\theta = \arcsin\left(\frac{x}{a}\right)
\]
le \(\sin 2\theta = 2\sin\theta\cos\theta = 2\frac{x}{a}\frac{\sqrt{a^2-x^2}}{a}=\frac{2x\sqrt{a^2-x^2}}{a^2}\).
Sephetho sa ho qetela:
\[
\int \sqrt{a^2-x^2}\,dx
= \frac{a^2}{2}\arcsin\left(\frac{x}{a}\right)+\frac{x}{2}\sqrt{a^2-x^2}+C
\]
Sebopeho sena se atisa ho hlaha mathateng a sebaka sa selikalikoe kapa jiometri.
5. Mohlala oa 2: Karolo e kopaneng ea foromo \(\sqrt{a^2+x^2}\)
Nahana ka:
\[
\int \sqrt{x^2+a^2}\,dx
\]
Phetolo:
\[
x = a\tan\theta,\quad dx = a\sec^2\theta\,d\theta
\]
Kahoo:
\[
\sqrt{x^2+a^2}=\sqrt{a^2\tan^2\theta+a^2}=a\sec\theta
\]
Karolo e kopaneng e ba:
\[
\int (a\sec\theta)(a\sec^2\theta)\,d\theta = a^2\int \sec^3\theta\,d\theta
\]
Karolo e kopaneng ya \(\int \sec^3\theta d\theta\) e na le foromo ya kgale:
\[
\int \sec^3\theta\,d\theta = \frac{1}{2}\sec\theta\tan\theta+\frac{1}{2}\ln|\sec\theta+\tan\theta|+C
\]
E le hore:
\[
\int \sqrt{x^2+a^2}\,dx = \frac{a^2}{2}\sec\theta\tan\theta+\frac{a^2}{2}\ln|\sec\theta+\tan\theta|+C
\]
Khutlela ho \(x\). Kaha \(x=a\tan\theta\), joale \(\tan\theta = x/a\) le \(\sec\theta=\sqrt{1+\tan^2\theta}=\sqrt{1+x^2/a^2}=\frac{\sqrt{x^2+a^2}}{a}\).
Liphetho:
\[
\int \sqrt{x^2+a^2}\,dx
= \frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln\left|\frac{\sqrt{x^2+a^2}+x}{a}\right|+C
\]
Sena se hlaha hangata fisiks le palong ea bolelele ba likhutlo.
6. Malebela a bohlokoa a ho qoba liphoso
1. Khetha sebaka ho latela sebopeho sa radical. U se ke ua sebelisa \(x=a\sin\theta\) bakeng sa \(\sqrt{a^2+x^2}\), hobane boitsebiso ha bo tšoane.
2. Ela hloko sebaka seo. Mohlala, ha o sebedisa \(\theta=\arcsin(x/a)\), hangata \(x\) e tlameha ho ba ho \([-a,a]\) hore metso e be ya nnete.
3. Sebedisa kgutlotharo e thusang ho kgutlela ho \(x\).
Mohlala: haeba \(x=a\tan\theta\), haha khutlotharo e nepahetseng e nang le lehlakore le fapaneng \(x\), lehlakore le haufi \(a\), e le hore hypotenuse e be \(\sqrt{x^2+a^2}\). Ho tloha moo \(\sec\theta = \frac{\sqrt{x^2+a^2}}{a}\), \(\sin\theta=\frac{x}{\sqrt{x^2+a^2}}\), jj.
4. Eba makhethe ha o nkela sebaka \(dx\). Liphoso tse ngata li hlaha ka lebaka la ho lebala ho fetola phapang.
7. Qetello
Phetolo ea Trigonometric ke mokhoa o thusang haholo bakeng sa ho rarolla li-integral tse kenyeletsang motso o sekoere oa polelo ea quadratic. Ka ho lemoha mekhoa e meraro e meholo—\(\sqrt{a^2-x^2}\), \(\sqrt{a^2+x^2}\), le \(\sqrt{x^2-a^2}\)—re ka khetha phetolo e nepahetseng 'me ra nolofatsa karolo e kopaneng ka mokhoa o hlophisehileng. Le hoja mehato e ka bonahala e le telele, mokhoa o tsitsitseng o tla etsa hore ts'ebetso e iketsahalle feela ebile e be bonolo. Qetellong, phetolo ea trigonometric ha se "leqheka" feela, empa ke leano la lipalo le sebelisang matla a boitsebiso ba trigonometric ho rarolla mathata a rarahaneng a kopaneng.
Haeba o lakatsa, nka eketsa karolo e khethehileng e nang le lipotso tsa ho itloaetsa (hammoho le lipuisano) e le hore sengoloa sena le sona se lokele ho sebelisoa e le thepa ea ho ruta.
