Ex 7.8, 8 - Chapter 7 Class 12 Integrals
Last updated at April 16, 2024 by Teachoo
Ex 7.8
Ex 7.8, 2
Ex 7.8, 3
Ex 7.8, 4 Important
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Ex 7.8, 6
Ex 7.8, 7
Ex 7.8, 8 Important You are here
Ex 7.8, 9
Ex 7.8, 10
Ex 7.8, 11 Important
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Ex 7.8, 14 Important
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Ex 7.8, 16 Important
Ex 7.8, 17 Important
Ex 7.8, 18
Ex 7.8, 19 Important
Ex 7.8, 20 Important
Ex 7.8, 21 (MCQ) Important
Ex 7.8, 22 (MCQ)
Last updated at April 16, 2024 by Teachoo
Ex 7.8, 8 ∫_(𝜋/6)^(𝜋/4)▒〖𝑐𝑜𝑠𝑒𝑐 𝑥〗𝑑𝑥 Let F(𝑥)=∫1▒〖𝑐𝑜𝑠𝑒𝑐 𝑥 . 𝑑𝑥〗 Multiplying and Dividing by 𝑐𝑜𝑠𝑒𝑐 𝑥+𝑐𝑜𝑡 𝑥 F(𝑥)=∫1▒(𝑐𝑜𝑠𝑒𝑐 𝑥 (𝑐𝑜𝑠𝑒𝑐 𝑥 + 𝑐𝑜𝑡 𝑥))/(𝑐𝑜𝑠𝑒𝑐 𝑥 + 𝑐𝑜𝑡 𝑥) 𝑑𝑥 Let c𝑜𝑠𝑒𝑐 𝑥+𝑐𝑜𝑡 𝑥=𝑡 Differentiating w.r.t. 𝑥 𝑑/𝑑𝑥 (𝑐𝑜𝑠𝑒𝑐 𝑥+𝑐𝑜𝑡 𝑥)=𝑑𝑡/𝑑𝑥 −𝑐𝑜𝑠𝑒𝑐^2 𝑥−𝑐𝑜𝑠𝑒𝑐 𝑥 𝑐𝑜𝑡 𝑥=𝑑𝑡/𝑑𝑥 −𝑐𝑜𝑠𝑒𝑐 𝑥(𝑐𝑜𝑠𝑒𝑐 𝑥+𝑐𝑜𝑡 𝑥)=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡/(−𝑐𝑜𝑠𝑒𝑐 𝑥 (𝑐𝑜𝑠𝑒𝑐 𝑥 + 𝑐𝑜𝑡 𝑥) ) Therefore, ∫1▒(𝑐𝑜𝑠𝑒𝑐 𝑥(𝑐𝑜𝑠𝑒𝑐 𝑥 + 𝑐𝑜𝑡 𝑥))/(𝑐𝑜𝑠𝑒𝑐 𝑥 + 𝑐𝑜𝑡 𝑥) 𝑑𝑥 =∫1▒(𝑐𝑜𝑠𝑒𝑐 𝑥(𝑐𝑜𝑠𝑒𝑐 𝑥 + 𝑐𝑜𝑡 𝑥))/𝑡. 𝑑𝑡/(−𝑐𝑜𝑠𝑒𝑐 𝑥(𝑐𝑜𝑠𝑒𝑐 𝑥 + 𝑐𝑜𝑡 𝑥) ) =−∫1▒𝑑𝑡/𝑡 =−log〖 |𝑡|〗 =−log〖 |𝑐𝑜𝑠𝑒𝑐 𝑥+𝑐𝑜𝑡 𝑥|〗 Hence, F(𝑥)=−log|𝑐𝑜𝑠𝑒𝑐 𝑥+cot𝑥 | Now, ∫_(𝜋/6)^(𝜋/4)▒〖𝑐𝑜𝑠𝑒𝑐 𝑥=𝐹(𝜋/4)−𝐹(𝜋/6) 〗 =−𝑙𝑜𝑔|𝑐𝑜𝑠𝑒𝑐(𝜋/4)+𝑐𝑜𝑡(𝜋/4)|− (−log|𝑐𝑜𝑠𝑒𝑐(𝜋/6)+ 𝑐𝑜𝑡(𝜋/6)| ) =−𝑙𝑜𝑔|√2+1|+𝑙𝑜𝑔|2+√3| =−𝑙𝑜𝑔|2+√3|− 𝑙𝑜𝑔|√2+1| =𝑙𝑜𝑔|(2 + √3)/(√2 + 1)| =𝑙𝑜𝑔|(2 + √3)/(√2 + 1)×(2 − √3)/(2 − √3)| =𝑙𝑜𝑔[((2)^2 − (√3)^2)/((√2 + 1) × (2 − √3) )] =𝑙𝑜𝑔[(4 − 3)/(√2 + 1)(2 − √3) ] =𝑙𝑜𝑔[(1 × (√2 − 1))/((2 − √3)(√2 + 1) ×(√2 − 1) )] =𝑙𝑜𝑔[(√2 − 1)/(2 − √3)[(√2)^2− 1^2 ] ] =𝒍𝒐𝒈[(√𝟐 − 𝟏)/(𝟐 − √𝟑)]