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Solution to a trigonometric problem using Weierstrass substitution

B.Sudhakar
02/01/2017 0 1

Let us solve one problem using Weierstrass substitution

∫ 1/(1 + cos x + sin x) dx     applying Weierstrass substitution

t= tan x/2 , sin x = 2 t/(1 + t²), cos x= 1 - t²)/ (1 + t²) , dx= 2 dt/ (1 + t²)

= ∫ 1/ 1 +( 1 - t²/1 + t²) + (2 t / 1 + t²) • 2/1 + t² dt 

= 2 ∫ 1/ 1 + 1 - t²/1 + t² + 2 t /1 + t² • 1/1 + t² dt 

= 2 ∫ 1/ [ (1 + t² + 1 - t² + 2 t) / (1 + t²) ] • (1/1 + t²) dt { take common denominator}

= 2 ∫ 1/ [ (2 + 2 t) / (1 + t²) ] • 1/(1 + t²) dt 

= 2 ∫ (1 + t²) / (2 + 2 t) • ( 1 + t² ) dt                { (1 + t²) cancels out }

= 2 ∫ 1 / (2 + 2 t) dt 

= 2 ∫ 1 / 2 ( 1 + t ) dt 

= 2/2 ∫ 1/ ( 1 + t ) dt 

= ∫ 1/ ( 1 + t ) dt 

= ln | 1 + t | dt + C                                              { substituting back t = tan x/2 }

= ln | 1 + tan x/2 | + C 

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Harsha | 07/01/2017

Unknown to me ? Never heard of it. Quite useful

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