Question 19

Find: ∫ (x 4   + 1) / x (x 2   + 1) 2  dx

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  1. Class 12
  2. Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Transcript

Question 19 Find: ∫1▒(𝑥^4 + 1)/(𝑥(𝑥^2 + 1)^2 ) dx ∫1▒(𝑥^4 + 1)/(𝑥(𝑥^2 + 1)^2 ) dx Writing x4 +1 = x4 + 1 + 2x2 – 2x2 ∫1▒(𝑥^4 + 1)/(𝑥(𝑥^2 + 1)^2 ) dx = ∫1▒(𝑥^4 + 1 + 2𝑥^2 − 2𝑥^2)/(𝑥(𝑥^2 + 1)^2 ) dx = ∫1▒((𝑥^2 + 1)^2 − 2𝑥^2)/(𝑥(𝑥^2 + 1)^2 ) dx = ∫1▒(𝑥^2 + 1)^2/(𝑥(𝑥^2 + 1)^2 ) dx – ∫1▒(2𝑥^2)/(𝑥(𝑥^2 + 1)^2 ) dx = ∫1▒1/𝑥 dx – ∫1▒2𝑥/(𝑥^2 + 1)^2 dx = log⁡〖|𝑥|〗 – ∫1▒2𝑥/(𝑥^2 + 1)^2 dx Let x2 + 1 = t 2x dx = dt = log⁡〖|𝑥|〗 – ∫1▒𝑑𝑡/𝑡^2 = log⁡〖|𝑥|〗 – 𝑡^(−2 + 1)/(−2 + 1) + C = log⁡〖|𝑥|〗 – 𝑡^(−1)/(−1) + C = log⁡〖|𝑥|〗 + 1/t + C Putting back t = x2 + 1 = log⁡〖|𝑥|〗 + 1/(x^2 + 1) + C

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.