Name: using substitution to evaluate definite integrals
Uploaded: 2026-01-12T12:24:59.000Z
Duration: 1 min 24 s
Channel: JoVE

Argümanı F(g(x)) bileşik fonksiyonunun zincir kuralı türevi olarak yazılabilen bir integralı ele alalım. İntegral çözüm için süreç zincir kuralının farklılaştırılmasını tersine çevirmeyi içerir.Burada, yeni bir değişken olan u, g(x) olarak tanımlanır. Sonra u'yu x ile karşılaştırın. Bu, du cinsinden yeniden düzenlenebilir.Sınırları değiştirmek için, x=a eşit g(a) yeni alt sınır olduğunda u eşit g(a) yeni alt sınır olur ve x=b olduğunda u eşit g(b) yeni üst sınır olur.Orijinal integral, u ve du yerine koyarak ve a ile b limitleri sırasıyla g(a) ve g(b) ile değiştirilerek yeniden yazılır. Yeni integrandı U'ya göre entegre etmek ve değişen sınırları uygulamak, nihai sayısal ifadeyi verir.Bir ikame örneği elektrik mühendisliğindedir; burada belirli bir zaman aralığında devreden toplam yükü bulmak için kullanılır. Burada toplam yük, zamana göre belirli integral hesaplanarak bulunur. Akım karmaşık bir fonksiyonla verildiğinden, yerine koyma integralı daha kolay çözülür.

belirli İntegrallere uygulanan yerine koyma kuralı

&#x130;ntegrand&#x131;n&#x131;n bile&#x15F;ik bir fonksiyonun yap&#x131;s&#x131;na uydu&#x11F;u belirli bir integrali de&#x11F;erlendirirken, yerine koyma y&#xF6;ntemi hesaplamay&#x131; basitle&#x15F;tirmek i&#xE7;in etkili bir yol sunar. Bu y&#xF6;ntem, t&#xFC;rev alma i&#x15F;lemindeki zincir kural&#x131;n&#x131;n tersine uygulanmas&#x131;na dayan&#x131;r ve karma&#x15F;&#x131;k bir ifadenin daha yal&#x131;n bir bi&#xE7;imde yeniden yaz&#x131;lmas&#x131;na olanak tan&#x131;r. &#x130;ntegrand bir i&#xE7; fonksiyon ile onun t&#xFC;revini i&#xE7;erdi&#x11F;inde, yerine koyma y&#xF6;ntemi problemin karma&#x15F;&#x131;kl&#x131;&#x11F;&#x131;n&#x131; do&#x11F;al olarak azalt&#x131;r.Belirli integrallerde yerine koyma y&#xF6;nteminin temel fikri, a&#x15F;a&#x11F;&#x131;da g&#xF6;sterilen ili&#x15F;kiyle &#xF6;zetlenir.\begin{equation*}\int_a^b f(g(x))\,g&#x27;(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du\\end{equation*}S&#xFC;re&#xE7;, i&#xE7; i&#x15F;levin belirlenmesi ve bunu temsil eden yeni bir de&#x11F;i&#x15F;kenin tan&#x131;mlanmas&#x131;yla ba&#x15F;lar. Tan&#x131;mlanan bu yeni de&#x11F;i&#x15F;kenin t&#xFC;revi al&#x131;narak, orijinal diferansiyelin yerini alan yeni bir diferansiyel elde edilir. Ayn&#x131; zamanda, integrasyon s&#x131;n&#x131;rlar&#x131; i&#xE7; fonksiyonun ba&#x15F;lang&#x131;&#xE7; ve biti&#x15F; noktalar&#x131;ndaki de&#x11F;erleri hesaplanarak g&#xFC;ncellenir. Bu i&#x15F;lem, t&#xFC;m integralin yeni de&#x11F;i&#x15F;ken cinsinden tutarl&#x131; bir bi&#xE7;imde ifade edilmesini sa&#x11F;lar ve integral al&#x131;nd&#x131;ktan sonra tekrar orijinal de&#x11F;i&#x15F;kene geri d&#xF6;nme gereklili&#x11F;ini ortadan kald&#x131;r&#x131;r.Yerine koyma i&#x15F;lemi tamamland&#x131;&#x11F;&#x131;nda, integrali de&#x11F;erlendirmek daha kolay h&#xE2;le gelir; &#xE7;&#xFC;nk&#xFC; integrand art&#x131;k tek bir de&#x11F;i&#x15F;kene ba&#x11F;l&#x131; hale gelir. G&#xFC;ncellenmi&#x15F; s&#x131;n&#x131;rlar&#x131;n do&#x11F;rudan uygulanmas&#x131;, integralin son say&#x131;sal de&#x11F;erini verir. Bu yakla&#x15F;&#x131;m, standart integral alma teknikleriyle &#xE7;&#xF6;z&#xFC;lmesi zor veya pratik olmayan hesaplamalar&#x131; &#xF6;nemli &#xF6;l&#xE7;&#xFC;de kolayla&#x15F;t&#x131;r&#x131;r.Yerine koyma y&#xF6;ntemi, elektrik m&#xFC;hendisli&#x11F;i gibi &#xE7;e&#x15F;itli uygulamal&#x131; alanlarda &#xF6;nemli bir rol oynar. &#xD6;rne&#x11F;in, belirli bir zaman aral&#x131;&#x11F;&#x131;nda bir devreden ge&#xE7;en toplam elektrik y&#xFC;k&#xFC;n&#xFC;n hesaplanmas&#x131;, bu s&#xFC;re boyunca akan ak&#x131;m&#x131;n zaman aral&#x131;&#x11F;&#x131;nda toplanmas&#x131;n&#x131; gerektirir. Ak&#x131;m karma&#x15F;&#x131;k bir &#x15F;ekilde tan&#x131;mland&#x131;&#x11F;&#x131;nda, yerine koyma y&#xF6;ntemi integralin verimli bir bi&#xE7;imde hesaplanmas&#x131;na imk&#xE2;n tan&#x131;r ve bu &#xF6;zelli&#x11F;iyle ger&#xE7;ek d&#xFC;nya sistemlerinin analizinde pratik ve g&#xFC;&#xE7;l&#xFC; bir ara&#xE7; olarak &#xF6;ne &#xE7;&#x131;kar.

Belirli İntegrallere Uygulanan Yerine Koyma Kuralı

&#x130;ntegrand&#x131;n&#x131;n bile&#x15F;ik bir fonksiyonun yap&#x131;s&#x131;na uydu&#x11F;u belirli bir integrali de&#x11F;erlendirirken, yerine koyma ...

substitution-rule-applied-to-definite-integrals

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Integrals

İntegraller

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

Substitution Rule Applied to Definite Integrals

using substitution to evaluate definite integrals

Substitution makes definite integrals easier to evaluate when the integrand is a composite function. It works by reversing the chain rule from differentiation. When the expression includes an inner function and its derivative, substitution can simplify the setup quickly.

The method starts by choosing the inner function and naming it with a new variable. Then its differential is found and used to replace the original variable terms. The limits of integration are also changed by plugging the original endpoints into the inner function. This keeps the integral written in one variable from start to finish.

After the change of variables, the integral usually becomes simpler to solve. Because the updated limits are used directly, there is no need to convert back to the original variable at the end. This makes the final numerical answer easier to obtain.

Substitution is useful when standard integration would be difficult or slow. It is also important in applied science and engineering. For example, in electrical engineering, it can help find the total electric charge that flows through a circuit over a time interval by integrating the current. If the current is given by a complicated expression, substitution makes the calculation more practical.

Using Substitution to Evaluate Definite Integrals

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substitution rule applied to definite integrals

Substitution makes definite integrals easier to evaluate when the integrand is a composite function. It works by reversing the chain rule from differentiation. When the expression includes an inner function and its derivative, substitution can simplify the setup quickly. The method starts by choosing the inner function and naming it with a new variable. Then its differential is found and used to replace the original variable terms. The limits of integration are also changed by plugging the...

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key terms and concepts

Learn how substitution simplifies definite integrals by changing variables, updating limits, and solving the integral directly. Includes a circuit current example.

Using Substitution to Evaluate Definite Integrals | JoVE K12

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calculus

Using Substitution to Evaluate Definite Integrals

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