Ex 7.5
Ex 7.5, 2
Ex 7.5, 3 Important
Ex 7.5, 4
Ex 7.5, 5
Ex 7.5, 6 Important
Ex 7.5, 7 Important
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Ex 7.5, 9 Important
Ex 7.5, 10
Ex 7.5, 11 Important
Ex 7.5, 12
Ex 7.5, 13 Important
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Ex 7.5, 16 Important
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Ex 7.5, 18 Important
Ex 7.5, 19
Ex 7.5, 20 Important
Ex 7.5, 21 Important
Ex 7.5, 22 (MCQ)
Ex 7.5, 23 (MCQ) Important
Last updated at April 16, 2024 by Teachoo
Ex 7.5, 1 𝑥/((𝑥 + 1) (𝑥 + 2) ) Solving integrand 𝑥/((𝑥 + 1) (𝑥 + 2) ) We can write it as 𝒙/((𝒙 + 𝟏) (𝒙 + 𝟐) ) " " = 𝑨/((𝒙 + 𝟏) ) + 𝑩/((𝒙 + 𝟐) ) 𝑥/((𝑥 + 1) (𝑥 + 2) ) " " = (𝐴(𝑥 + 2) + 𝐵 (𝑥 + 1))/((𝑥 + 1) (𝑥 + 2) ) Cancelling denominator 𝒙 = 𝑨(𝒙+𝟐) + 𝑩(𝒙+𝟏) Putting 𝒙=−𝟏 in (1) −1=𝐴(−1+2) + 𝐵(−1+1) −1=𝐴 × 1+ 𝐵 × 0 −1=𝐴 𝑨=−𝟏 Putting 𝒙=−𝟐 in (1) −2 = 𝐴(−2+2) + 𝐵(−2+1) −2 = 𝐴 × 0+ 𝐵 × (−1) −2 = − 𝐵 𝑩 = 𝟐 Hence we can write it as 𝒙/((𝒙 + 𝟏) (𝒙 + 𝟐) ) = (−𝟏)/((𝒙 + 𝟏) ) + 𝟐/((𝒙 + 𝟐) ) Therefore ∫1▒𝒙/((𝒙 + 𝟏) (𝒙 + 𝟐) ) 𝒅𝒙 = ∫1▒(−1)/((𝑥 + 1) ) 𝑑𝑥 + ∫1▒2/((𝑥 + 2) ) 𝑑𝑥 = −1∫1▒1/((𝑥 + 1) ) 𝑑𝑥 + 2∫1▒1/((𝑥 + 2) ) 𝑑𝑥 = −〖𝐥𝐨𝐠 〗|𝒙+𝟏|+𝟐 〖𝐥𝐨𝐠 〗|𝒙+𝟐|+𝑪 = −〖log 〗|𝑥+1|+〖log 〗〖|𝑥+2|^2 〗+𝐶 = 〖log 〗|(𝑥 + 2)^2/(𝑥 + 1)|+𝐶 = 〖𝐥𝐨𝐠 〗〖(𝒙 + 𝟐)^𝟐/|𝒙 + 𝟏| 〗 +𝑪 As 𝒍𝒐𝒈 𝑨−𝒍𝒐𝒈 𝑩=𝑙𝑜𝑔 𝐴/𝐵 As (𝑥+2)^2is always positive