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What is Weierstrass substitution in integration ?

B.Sudhakar
01/01/2017 0 2

Weierstrass substitution in integration

put t = tan (x/2)---------------------A

sin x = 2 t / (1 + t²)----------------B

cos x = (1 - t²) / (1 + t²)---------C

dx=2 dt / (1+ t²)---------------------D

When t= tan x/2 , sin x= ? cos x=?  dx=? All these doubts are cleared below 

 

dt/dx (tan x/2)  = sec²( x/2) • (1/2)= (1/2) • (√t² + 1)² = (t² + 1)/2

                                                                                                        { dy/dx tan x = sec² x }

∴ dx= dt/(t² + 1)/2 = 2 dt/(t² + 1)---------------------------D

As  tan x/2 = t, in the right angle triangle, opp side= t, adj side = 1 and hyp =(√t²+1)

∴sin x/2 = t/√t²+1 and cos x/2 = 1/√t²+1

We now have to convert sin x/2 to sin x and cos x/2 to cos x

sin x = sin ( 2 • x/2 ) = 2 sin x/2 • cos x/2                               { sin 2x = 2 sin x cos x )

= 2 ( t/√t² + 1 ) (1/√t² + 1 ) = 2 t / (t² + 1)

∴ sin x = 2 t/(t² + 1)------------------------------------------------B

cos x = cos ( 2 • x/2 )= 2 cos² x/2 - 1                                        { cos 2x = 2 cos²x — 1 }

= 2 { (1/√t² + 1) }² —1 

= 2/ ( t² + 1 ) — 1= 2 —( t² + 1 )/(t² + 1 )

= (1 - t²)/(1 + t²)

∴ cos x= (1— t²)/(1 + t²)-----------------------------------------C

 

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K

Krithika | 22/01/2017

Excellent lesson.

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K

Karthik | 04/01/2017

This method is not known to many. Thanks a lot for uploading the lesson.

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