Dažādi jautājumi matemātikā.

[Pauchuk]

Sveiki. Lai nav jācepj jauns topiks par katru sīku jautājumu, varbūt varētu izmantot šo sadaļu.

Vai varētu kāds paskaidrot, kā "panest aiz diferenciāļa", kad nepieciešams integrēt, process notiek, vienkāršiem vārdiem? Vai kā atrast youtube?

[joo]

Aplūkosim piemēru: int 2x * cos(x^2) dx.

Te varētu iztikt ar substitūciju: u=x^2. Tad du = 2x dx (atvasinām abas substitūcijas puses pēc x) un integrālis top par: int cos(u) du, ko var nointegrēt vieglāk.

Tikpat labi varējām ievērot, ka d/dx (x^2) = 2x jeb d(x^2) = 2x dx (vai otrā virzienā: 2x dx -> d(x^2) ). Šo sauc par panešanu zem diferenciāļa. Tad mūsu jaunais integrēšanas mainīgais ir nevis x, bet x^2, ko ērtības labad varam nosaukt par u.

Esam ieguvuši jaunu integrāli: int cos(u) du.

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Stāsta morāle: "panest zem diferenciāļa" tā pati substitūcija būtībā vien ir.
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Panešanas galvenais jautājums ir: "kas man būtu jāatvasina, lai iegūtu to, ko es gribu panest zem diferenciāļa ?"
Random piemēri panešanai: x^2 dx = d(x^3/3); sin(x) dx = d(-cos(x)); 1/x dx = d(ln(x))

[Pauchuk]

Tā īsumā - to, ko liek pie dx - x vietā, to atvasina un liek apakšā?

[joo]

Integrē un nes apakšā, jo "d" ir atvasinātājs. Tas atvasina to, kas sēž aiz tā:

x dx -> 1/2 d(x^2)

[Pauchuk]

Te redzams, ka dx (x vietā ir izteiksme) un apakšā nāk klāt atvasinājums, tad to visu integrē..

[joo]

Ideja ir panešanu zem diferenciāļa ir tāda, ka šī darbība nemaina izteiksmes vērtību. Tāpat kā vienādojuma abu pušu reizināšana ar skaitli, piemēram.

Attiecīgi, ja mums sākumā ir tikai "dx" , mēs varam to aizstāt ar 1/2 * d(2x - 1), jo tas ir tas pats (atceries, ka "d" ir atvasināšanas operators):

1/2 * d(2x - 1) = 1/2 * (2 * dx - 0) = 2/2 dx = dx.

Šādas mahinācijas varam darīt neatkarīgi no pārējās integrāļa izteiksmes, vienkārši mērķis ir zem "d" iemānīt tādu izteiksmi, kura parādās arī pašā integrālī, jo tad varam pa taisno nointegrēt (joprojām tas pats princips, kas ar substitūciju).

[Pauchuk]

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