How do you calculate ∫ 0 5 π / 2 d x 2 + cos...

gaiaecologicaq2

gaiaecologicaq2

Answered

2022-07-01

How do you calculate
0 5 π / 2 d x 2 + cos x
I tried all available substitutions including tangent half angle, but all these substitutions do not distinguish between π / 2 and 5 π / 2. I tried splitting up the integral into two parts, with one part from 0 to 2 π, but again this had problems as 0 and 2 π were basically "the same"

Answer & Explanation

1s1zubv

1s1zubv

Expert

2022-07-02Added 17 answers

Substitute (Weierstrass):
t = tan x 2 cos x = 1 t 2 1 + t 2 , d x = 2 1 + t 2 d t
by the period of cosx (and also of tanx , in fact), we get
0 5 π / 2 = 0 π d x 2 + cos x + π 2 π d x 2 + cos x + 2 π 5 π / 2 d x 2 + cos x =
= 0 2 d t ( 1 + t 2 ) ( 2 + 1 t 2 1 + t 2 ) + 0 2 d t ( 1 + t 2 ) ( 2 + 1 t 2 1 + t 2 ) + 0 1 2 d t ( 1 + t 2 ) ( 2 + 1 t 2 1 + t 2 ) =
= 2 d t 3 + t 2 + 2 0 1 d t 3 + t 2 = 2 3 ( 1 3 d t 1 + ( t 3 ) 2 ) + 2 3 ( 0 1 1 3 d t 1 + ( t 3 ) 2 ) =
= 2 3 arctan t 3 | + 2 3 arctan t 3 | 0 1 = 2 3 ( π + π 6 ) = 7 π 3 3

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