Ex 7.3, 20 - Chapter 7 Class 12 Integrals
Last updated at April 16, 2024 by Teachoo
Integration using trigo identities - 2x formulae
Integration using trigo identities - 2x formulae
Last updated at April 16, 2024 by Teachoo
Ex 7.3, 20 Integrate the function cos2𝑥/(cos〖𝑥 〗+ sin𝑥 )^2 ∫1▒cos2𝑥/(cos𝑥 + sin𝑥 )^2 =∫1▒(cos^2𝑥 − sin^2𝑥)/(cos𝑥 + sin𝑥 )^2 𝑑𝑥 =∫1▒(cos𝑥 − sin𝑥 )(cos𝑥 + sin𝑥 )/(cos𝑥 + sin𝑥 )^2 𝑑𝑥 =∫1▒(cos𝑥 − sin𝑥)/(cos𝑥 + sin𝑥 ) 𝑑𝑥 Let cos𝑥+sin𝑥=𝑡 Differentiating w.r.t. x (𝑐𝑜𝑠2𝜃=〖𝑐𝑜𝑠〗^2𝜃−〖𝑠𝑖𝑛〗^2𝜃) −sin𝑥+cos𝑥=𝑑𝑡/𝑑𝑥 (cos𝑥−sin𝑥 )𝑑𝑥=𝑑𝑡 𝑑𝑥=1/((cos𝑥 − sin𝑥 ) ) 𝑑𝑡 Thus, our equation becomes =∫1▒((cos𝑥 − sin𝑥 ))/𝑡 ×𝑑𝑡/(cos𝑥 − sin𝑥 ) =∫1▒1/𝑡 𝑑𝑡 =log|𝑡|+𝐶 Putting value of 𝑡=𝑐𝑜𝑠𝑥+𝑠𝑖𝑛𝑥 =𝒍𝒐𝒈|𝒄𝒐𝒔𝒙+𝒔𝒊𝒏𝒙 |+𝑪