Question 19

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

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Question 19 Find: โˆซ1โ–’(๐‘ฅ^4 + 1)/(๐‘ฅ(๐‘ฅ^2 + 1)^2 ) dx โˆซ1โ–’(๐‘ฅ^4 + 1)/(๐‘ฅ(๐‘ฅ^2 + 1)^2 ) dx Writing x4 +1 = x4 + 1 + 2x2 โ€“ 2x2 โˆซ1โ–’(๐‘ฅ^4 + 1)/(๐‘ฅ(๐‘ฅ^2 + 1)^2 ) dx = โˆซ1โ–’(๐‘ฅ^4 + 1 + 2๐‘ฅ^2 โˆ’ 2๐‘ฅ^2)/(๐‘ฅ(๐‘ฅ^2 + 1)^2 ) dx = โˆซ1โ–’((๐‘ฅ^2 + 1)^2 โˆ’ 2๐‘ฅ^2)/(๐‘ฅ(๐‘ฅ^2 + 1)^2 ) dx = โˆซ1โ–’(๐‘ฅ^2 + 1)^2/(๐‘ฅ(๐‘ฅ^2 + 1)^2 ) dx โ€“ โˆซ1โ–’(2๐‘ฅ^2)/(๐‘ฅ(๐‘ฅ^2 + 1)^2 ) dx = โˆซ1โ–’1/๐‘ฅ dx โ€“ โˆซ1โ–’2๐‘ฅ/(๐‘ฅ^2 + 1)^2 dx = logโกใ€–|๐‘ฅ|ใ€— โ€“ โˆซ1โ–’2๐‘ฅ/(๐‘ฅ^2 + 1)^2 dx Let x2 + 1 = t 2x dx = dt = logโกใ€–|๐‘ฅ|ใ€— โ€“ โˆซ1โ–’๐‘‘๐‘ก/๐‘ก^2 = logโกใ€–|๐‘ฅ|ใ€— โ€“ ๐‘ก^(โˆ’2 + 1)/(โˆ’2 + 1) + C = logโกใ€–|๐‘ฅ|ใ€— โ€“ ๐‘ก^(โˆ’1)/(โˆ’1) + C = logโกใ€–|๐‘ฅ|ใ€— + 1/t + C Putting back t = x2 + 1 = logโกใ€–|๐‘ฅ|ใ€— + 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 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.