Question 9
Find 1 ^ (1 + sin 2 )/(1 + cos 2 ) dx
1 ^ (1 + 2 )/(1 + 2 ) dx
Putting sin 2x = 2 sin x cos x
cos 2x = 2 cos2 x 1
= 1 ^ (1 + 2 sin cos )/(1 + (2 cos^2 1)) dx
= 1 ^ (1 + 2 sin cos )/(2 cos^2 ) dx
Putting 1 = sin2 x + cos2 x
= 1 ^ (sin^2 + cos^2 + 2 sin cos )/(2 cos^2 ) dx
= 1 (( + )^2 )/(2 2 )dx
= 1/2 1 ^ (sin /( 2 )+cos /( 2 )) = 1/2 1 ("sec sec tan x " )
= 1/2 ^ sec x + c [ ^ ( ( )+ ^ ( )) = ^ ( )+ ]
Using (a + b)2 = a2 + b2 + 2ab
where a = sin x , b = cos x
= 1 ^ (( + )^2 )/(2 2 ) dx
= 1/2 1 ^ (( + ))/ 2 dx
= 1/2 1 ^ (sin /( 2 )+cos /( 2 ))
= 1/2 1 ^ (sin /( ) 1/cos +1/cos )
= 1/2 1 ^ (tan sec +sec )
= 1/2 1 ^ (sec +tan sec )
Using 1 ^ ( ( )+ ^ ( )) = ^ ( )+
Where f(x) = sec x
= / ^ sec x + c

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.