The method of integration can be traced back to the Egyptians, in the Moscow Mathematical Papyrus circa 1800 BC, which gives the formula for finding the volume of a pyramidal frustrum.[1] Greek geometers are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create. Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes' Quadrature of the Parabola and Archimedes on Spheres & Cylinders.) It was not until the time of Newton that these methods were made obsolete. It should not be thought that infinitesimals were put on rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek Mathematicians accept a proposition as true. (See Archimedes' use of infinitesimals.) The method of exhaustion was rediscovered in China by Liu Hui in the 3rd century AD, who used it to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[1]

Integration by Substitution

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  • โจทย์ข้อนี้ใช้ Integration by substitution แบบที่ 1
  • สูตร
 
 
 
 
  • ค่าที่ต้องเก็บไว้ใช้ต่อคือ
 
  • ต่อไปก็ต้องเปลี่ยนสมการทั้งหมดให้อยู่ในรูป u
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  • ได้เฉย... ข้อนี้ง่ายซะงั้นอะ
  • แทนค่ากลับ
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Author: Surachai U.
Created Date: 2007-07-25 22:33:00
Last Update: 2007-07-25 23:33:00

Comment: 2

ถ้าคิดดีๆก็จะเท่ากันนะครับ
ถ้ากำหนด cos x = u
จะได้คือตอบเป็น -(cos^2 x)/2 +c

อาจจะงงว่าเท่ากันอย่างไร ให้ลองใช้สูตรนี้คิดต่อ 1=sin^2 x + cos^2 x ฉะนั้น sin^2 x = 1 - cos^2 x

เอา 2 หารตลอด (sin^2 x / 2) = 1/2 - (cos^2 x / 2) แต่ 1/2 ให้เอาไปรวมกับ c เพราะจัดว่าเป็นค่าคงที่

เพราะฉะนั้นจึงได้เท่ากันด้วยประการฉะนี้

Surachai U.
2008-08-05 11:30:00

Comment: 1

แล้วถ้าเอา Cos x เป็น u อ่ะคับ ผลออกมาไม่เหมือนกันนิคับ

mean
2008-08-05 01:00:58

In calculus, the substitution rule is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, the substitution rule is a relatively important tool for mathematicians. It is the counterpart to the chain rule of differentiation.