MATH1014_integration_I_Fall_2019_20 - Techniques of Integration Integration by parts(Mainly based on Stewart Chapter 7 \u00a77.1 Edmund Chiang MATH1014

MATH1014_integration_I_Fall_2019_20 - Techniques of...

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Techniques of Integration: Integration by parts (Mainly based on Stewart: Chapter 7: § 7.1) Edmund Chiang MATH1014 September 18, 2019 1 Integrations by recognition Unless one can find a primitive of a given function f ( x ) from a standardised integration table directly, one has to find other methods to achieve the goal. Table 1: An integration table Z x n dx = x n +1 n + 1 + C ( n 6 = 1) Z 1 x dx = ln | x | + C Z e x dx = e x + C Z a x dx = a x ln a dx + C Z sin x dx = - cos x + C Z cos x dx = sin x + C Z sec 2 x dx = tan x + C Z sec x tan x dx = sec x + C Z csc 2 x dx = - cot x + C Z sinh x dx = cosh x + C Z tan x dx = ln | sec x | + C Z cot x dx = ln | sin x | + C Z 1 x 2 + a 2 dx = 1 a tan - 1 ( x a ) + C Z 1 a 2 - x 2 dx = sin - 1 ( x a ) + C ( a > 0) R x b dt ( t 2 + a 2 )( t 2 - b 2 ) = 1 a 2 + b 2 cn - 1 ( b x , a a 2 + b 2 ) R x -∞ dt ( a - t )( b - t )( c - t ) = 2 a - c sn - 1 ( q a - c a - x , q a - b a - c ) where the functions sn( x, k ( a, b )) and cn( x, k ( a, b, c )) are called (Jacobian) elliptic functions and the two integrals are called elliptic integrals of the first kind 1 These elliptic functions 1 There are also the elliptic functions of the second and third kinds. (See DLMF) Neither of them are required in this course. 1

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