Example 33 - Chapter 7 Class 12 Integrals

Transcript

Example 33 (Method 1) Evaluate ∫_(𝜋/6)^(𝜋/3 )▒𝑑𝑥/(1 +√(tan⁡𝑥 )) Let I =∫_(𝜋/6)^(𝜋/3 )▒1/(1 + √(tan⁡𝑥 )).𝑑𝑥 I = ∫_(𝜋/6)^(𝜋/3 )▒1/(1 + √(sin⁡𝑥/cos⁡𝑥 )).𝑑𝑥 I = ∫_(𝜋/6)^(𝜋/3 )▒1/(1 + √(sin⁡𝑥 )/√(cos⁡𝑥 )).𝑑𝑥 I = ∫_(𝜋/6)^(𝜋/3 )▒1/( (√(cos⁡𝑥 ) + √(sin⁡𝑥 ))/√(cos⁡𝑥 )) 𝑑𝑥 I = ∫_(𝜋/6)^(𝜋/3 )▒(√(cos⁡𝑥 ) )/(√(cos⁡𝑥 ) + √(sin⁡𝑥 )) 𝑑𝑥 ∴ I =∫_(𝜋/6)^(𝜋/3 )▒(√(cos⁡〖 (𝜋/6 + 𝜋/3 − 𝑥)〗 ) )/(√(〖cos 〗⁡(𝜋/6 + 𝜋/3 − 𝑥) ) +√(〖sin 〗⁡(𝜋/6 + 𝜋/3 − 𝑥) )) 𝑑𝑥 I =∫_(𝜋/6)^(𝜋/3 )▒(√(cos⁡〖 (𝜋/2 − 𝑥)〗 ) )/(√(cos⁡〖 (𝜋/2 − 𝑥)〗 ) +√(sin⁡(𝜋/2 − 𝑥) )) 𝑑𝑥 I =∫_(𝜋/6)^(𝜋/3 )▒(√(sin⁡𝑥 ) )/(√(sin⁡𝑥 ) + √(cos⁡𝑥 )) 𝑑𝑥 Adding (1) and (2) i.e. (1) + (2) I + I = ∫_(𝜋/6)^(𝜋/3 )▒(√(cos 𝑥) )/(√(sin⁡𝑥 ) + √(cos⁡𝑥 )). 𝑑𝑥+∫_(𝜋/6)^(𝜋/3 )▒(√(sin⁡𝑥 ) )/(√(sin⁡𝑥 ) + √(cos⁡𝑥 )) 𝑑𝑥 Adding (1) and (2) i.e. (1) + (2) I + I = ∫_(𝜋/6)^(𝜋/3 )▒(√(cos 𝑥) )/(√(sin⁡𝑥 ) + √(cos⁡𝑥 )). 𝑑𝑥+∫_(𝜋/6)^(𝜋/3 )▒(√(sin⁡𝑥 ) )/(√(sin⁡𝑥 ) + √(cos⁡𝑥 )) 𝑑𝑥 2I =∫_(𝜋/6)^(𝜋/3 )▒(√(cos 𝑥) + √(sin⁡𝑥 ))/(√(cos 𝑥) + √(sin⁡𝑥 )) 𝑑𝑥 2I=∫_(𝜋/6)^(𝜋/3 )▒〖1.〗 𝑑𝑥 2I= [𝑥]_(𝜋/6)^(𝜋/3) I= 1/2 [𝜋/3−𝜋/6] I= 1/2 [(2𝜋 − 𝜋)/6] 𝐈= 𝝅/𝟏𝟐 Example 33 (Method 2) Evaluate ∫_(𝜋/6)^(𝜋/3 )▒𝑑𝑥/(1 +√(tan⁡𝑥 )) Let I =∫_(𝜋/6)^(𝜋/3 )▒1/(1 + √(tan⁡𝑥 )).𝑑𝑥 I = ∫_(𝜋/6)^(𝜋/3 )▒1/(1 + √(tan⁡(𝜋/3 + 𝜋/6 − 𝑥) )) 𝑑𝑥 I = ∫_(𝜋/6)^(𝜋/3 )▒1/(1 +√(tan⁡(𝜋/2 − 𝑥) )) 𝑑𝑥 I = ∫_(𝜋/6)^(𝜋/3 )▒(1 )/(1 + √(co𝑡⁡𝑥 ) ) 𝑑𝑥 Adding (1) and (2) I+I=∫_(𝜋/6)^(𝜋/3 )▒(( 1 )/( 1 + √(tan⁡𝑥 ))+1/(1 + √(cot⁡𝑥 ))) 𝑑𝑥 2I=∫_(𝜋/6)^(𝜋/3 )▒(( 1 + √(cot⁡𝑥 ) + 1 + √(tan⁡𝑥 ) ))/( 1 + √(tan 𝑥) )(1 + √(cot⁡𝑥 ) ) 𝑑𝑥 2I=∫_(𝜋/6)^(𝜋/3 )▒( 2 + √(cot⁡𝑥 ) + √(tan⁡𝑥 ))/( 1 + √(cot⁡𝑥 ) + √(tan⁡𝑥 ) + √(𝐭𝐚𝐧⁡〖𝒙 〗 ) . √(𝐜𝐨𝐭⁡𝒙 )) 𝑑𝑥 2I=∫_(𝜋/6)^(𝜋/3 )▒( 2 + √(cot⁡𝑥 ) + √(tan⁡𝑥 ))/( 1 + √(cot⁡𝑥 ) + √(tan⁡𝑥 ) + 𝟏) 𝑑𝑥 2I=∫_(𝜋/6)^(𝜋/3 )▒(2 + √(cot⁡𝑥 ) + √(tan⁡𝑥 ))/(2 + √(cot⁡𝑥 ) + √(tan⁡𝑥 )) 𝑑𝑥 2I=∫_(𝜋/6)^(𝜋/3 )▒𝑑𝑥 2I= [I]_(𝜋/6)^(𝜋/3) I= 1/2 [𝜋/3−𝜋/6] I= 1/2 [𝜋/6] 𝐈= 𝝅/𝟏𝟐