Question 9

Transcript

Question 9 Find ∫1▒〖𝑒^𝑥 √(1 + sin⁡2𝑥 )/(1 + cos⁡2𝑥 )〗 dx ∫1▒〖𝑒^𝑥 √(1 + 𝑠𝑖𝑛 2𝑥 )/(1 + 𝑐𝑜𝑠 2𝑥)〗 dx Putting sin 2x = 2 sin x cos x cos 2x = 2 cos2 x – 1 = ∫1▒〖𝑒^𝑥 √(1 + 2 sin⁡𝑥 cos⁡𝑥 )/(1 + (2 cos^2⁡𝑥 − 1))〗 dx = ∫1▒〖𝑒^𝑥 √(1 + 2 sin⁡𝑥 cos⁡𝑥 )/(2 cos^2⁡𝑥 )〗 dx Putting 1 = sin2 x + cos2 x = ∫1▒〖𝑒^𝑥 √(sin^2⁡𝑥 + cos^2⁡𝑥 + 2 sin⁡𝑥 cos⁡𝑥 )/(2 cos^2⁡𝑥 )〗 dx = ∫1▒√((𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥)^2 )/(2 𝑐𝑜𝑠2𝑥)dx = 1/2 ∫1▒〖𝑒^𝑥 (sin⁡𝑥/(𝑐𝑜𝑠2 𝑥)+cos⁡𝑥/(𝑐𝑜𝑠2 𝑥))𝑑𝑥〗 = 1/2 ∫1▒〖𝑒𝑥 ("sec sec tan x " )𝑑𝑥〗 = 1/2 𝑒^𝑥 sec x + c [∵𝑒^𝑥 (𝑓(𝑥)+𝑓^′ (𝑥))𝑑𝑥=𝑒^𝑥 𝑓(𝑥)+𝑐] Using (a + b)2 = a2 + b2 + 2ab where a = sin x , b = cos x = ∫1▒〖𝑒^𝑥 √((𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥)^2 )/(2 𝑐𝑜𝑠2𝑥)〗dx = 1/2 ∫1▒〖𝑒^𝑥 ((𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥))/𝑐𝑜𝑠2𝑥〗dx = 1/2 ∫1▒〖𝑒^𝑥 (sin⁡𝑥/(𝑐𝑜𝑠2 𝑥)+cos⁡𝑥/(𝑐𝑜𝑠2 𝑥))𝑑𝑥〗 = 1/2 ∫1▒〖𝑒^𝑥 (sin⁡𝑥/(𝑐𝑜𝑠 𝑥) ×1/cos⁡𝑥 +1/cos⁡𝑥 )𝑑𝑥〗 = 1/2 ∫1▒〖𝑒^𝑥 (tan⁡𝑥 sec⁡𝑥+sec⁡𝑥 )𝑑𝑥〗 = 1/2 ∫1▒〖𝑒^𝑥 (sec⁡𝑥+tan⁡𝑥 sec⁡𝑥 )𝑑𝑥〗 Using ∫1▒〖𝑒^𝑥 (𝑓(𝑥)+𝑓^′ (𝑥))𝑑𝑥〗=𝑒^𝑥 𝑓(𝑥)+𝑐 Where f(x) = sec x = 𝟏/𝟐 𝒆^𝒙 sec x + c