Ex 7.11, 12 - Class 12

Transcript

Ex 7.11, 12 By using the properties of definite integrals, evaluate the integrals: ∫_0^𝜋▒(𝑥 𝑑𝑥)/(1 + sin⁡𝑥 ) 𝑑𝑥 Let I=∫_0^𝜋▒𝑥/(1+ sin⁡𝑥 ) 𝑑𝑥 ∴ I=∫_0^𝜋▒(𝜋 − 𝑥)/(1+ sin⁡𝑥 ) 𝑑𝑥 Adding (1) and (2) i.e. (1) + (2) I+I=∫_0^𝜋▒( 𝑥)/(1+ sin⁡𝑥 ) 𝑑𝑥+∫_0^𝜋▒( 𝜋 − 𝑥)/(1+ sin⁡𝑥 ) 𝑑𝑥 2I=∫_0^𝜋▒( 𝑥 + 𝜋 − 𝑥)/(1+ sin⁡𝑥 ) 𝑑𝑥 2I=∫_0^𝜋▒( 𝜋)/(1+ sin⁡𝑥 ) 𝑑𝑥 I=𝜋/2 ∫_0^𝜋▒( 1)/(1+ sin⁡𝑥 ) 𝑑𝑥 Solving I(Method 1): I=𝜋/2 ∫_0^𝜋▒( 1)/(1 + sin⁡𝑥 ) 𝑑𝑥 Multiplying and dividing by cos⁡𝑥 ∴, I=𝜋/2 ∫_0^𝜋▒( 1/cos⁡𝑥 )/( (1+ sin⁡𝑥)/cos⁡𝑥 ) 𝑑𝑥 I=𝜋/2 ∫_0^𝜋▒sec⁡𝑥/( 1/〖 cos〗⁡〖𝑥 〗 + ( sin⁡𝑥)/cos⁡𝑥 ) 𝑑𝑥 I=𝜋/2 ∫_0^𝜋▒sec⁡𝑥/(sec⁡𝑥 + tan⁡𝑥 ) 𝑑𝑥 Let sec⁡𝑥+tan⁡〖𝑥=𝑡〗 Differentiating both sides w.r.t. 𝑥 sec⁡𝑥 tan⁡〖𝑥+sec^2⁡〖𝑥=𝑑𝑡/𝑑𝑥〗 〗 sec⁡𝑥 [tan⁡〖𝑥+sec⁡𝑥 〗 ]=𝑑𝑡/𝑑𝑥 sec⁡𝑥 [tan⁡〖𝑥+sec⁡𝑥 〗 ] 𝑑𝑥=𝑑𝑡 sec⁡𝑥 [𝑡] 𝑑𝑥=𝑑𝑡 𝑑𝑥=𝑑𝑡/(𝑡 sec⁡𝑥 ) Putting the values of dx and (sec⁡〖x+tan⁡x 〗 )=t in(3) I=𝜋/2 ∫_0^𝜋▒〖sec⁡𝑥/sec⁡〖𝑥 + tan⁡𝑥 〗 . 𝑑𝑥〗 I=𝜋/2 ∫_0^𝜋▒〖sec⁡𝑥/𝑡. 𝑑𝑥〗 I=𝜋/2 ∫_0^𝜋▒〖sec⁡𝑥/𝑡 ×𝑑𝑡/(𝑡 sec⁡𝑥 )〗 I=𝜋/2 ∫_0^𝜋▒〖1/𝑡^2 . 𝑑𝑡〗 I=𝜋/2 ∫_0^𝜋▒〖𝑡^(− 2) 𝑑𝑡〗 I= 〖𝜋/2 [𝑡^(− 2 + 1)/(− 2 + 1)]〗_0^𝜋 I= 〖𝜋/2 [𝑡^(−1)/(−1)]〗_0^𝜋 I=(− 𝜋)/2 [1/𝑡] I=(− 𝜋)/2 [1/sec⁡〖𝑥 + tan⁡𝑥 〗 ]_0^𝜋 I=(− 𝜋)/2 [1/sec⁡〖(𝜋) + tan⁡(𝜋) 〗 −1/sec⁡〖(0) + tan⁡(0) 〗 ] I=(− 𝜋)/2 [1/(− 1 + 0)− 1/( 1 + 0)] I=(− 𝜋)/2 [−1−1] I=(− 𝜋)/2 [− 2] I=𝜋 Solving I (Method 2): I=𝜋/2 ∫_0^𝜋▒( 1)/(1+ sin⁡𝑥 ) 𝑑𝑥 Multiplying and dividing by (1−sin⁡𝑥 ) ∴ I=𝜋/2 ∫_0^𝜋▒〖( 1)/(1+ sin⁡𝑥 ) × (1− sin⁡𝑥)/(1− sin⁡𝑥 )〗 . 𝑑𝑥 I=𝜋/2 ∫_0^𝜋▒(1− sin⁡𝑥)/(1− sin^2⁡𝑥 ) 𝑑𝑥 I=𝜋/2 ∫_0^𝜋▒(1− sin⁡𝑥)/( 〖cos 〗^2⁡𝑥 ) 𝑑𝑥 I=𝜋/2 ∫_0^𝜋▒[1/cos^2⁡𝑥 −sin⁡𝑥/( 〖cos 〗^2⁡𝑥 )] 𝑑𝑥 I=𝜋/2 ∫_0^𝜋▒[sec^2⁡𝑥−sin⁡𝑥/(cos⁡𝑥 .〖 cos〗⁡𝑥 )] 𝑑𝑥 I=𝜋/2 ∫_0^𝜋▒[sec^2⁡𝑥−tan⁡〖𝑥 sec⁡𝑥 〗 ] 𝑑𝑥 I=𝜋/2 [[tan⁡𝑥 ]_0^𝜋−[sec⁡𝑥 ]_0^𝜋 ] I=𝜋/2 [[𝑡𝑎𝑛(𝜋)−𝑡𝑎𝑛(0)]−[𝑠𝑒𝑐(𝜋)−𝑠𝑒𝑐(0)]] I=𝜋/2 [[0−0]−[−1−1]] I=𝜋/2 [0−(−2)] I=𝜋/2 [2] 𝐈=𝝅