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integral fraction numerator d t over denominator t to the power of 4 plus t cubed plus 2 t squared plus t plus 1 end fraction

Asked by Eng.surinder.singh | 26th Jan, 2019, 10:16: AM

Expert Answer:

begin mathsize 16px style straight I equals integral fraction numerator dt over denominator straight t to the power of 4 plus straight t cubed plus 2 straight t squared plus straight t plus 1 end fraction
Consider comma space
straight t to the power of 4 plus straight t cubed plus 2 straight t squared plus straight t plus 1
equals straight t to the power of 4 plus straight t cubed plus straight t squared plus straight t squared plus straight t plus 1
equals straight t squared open parentheses straight t squared plus straight t plus 1 close parentheses plus open parentheses straight t squared plus straight t plus 1 close parentheses
equals open parentheses straight t squared plus straight t plus 1 close parentheses open parentheses straight t squared plus 1 close parentheses
straight I equals integral fraction numerator dt over denominator open parentheses straight t squared plus straight t plus 1 close parentheses open parentheses straight t squared plus 1 close parentheses end fraction
Using space Partial space fraction
fraction numerator 1 over denominator open parentheses straight t squared plus straight t plus 1 close parentheses open parentheses straight t squared plus 1 close parentheses end fraction equals fraction numerator straight A over denominator straight t squared plus straight t plus 1 end fraction plus fraction numerator straight B over denominator straight t squared plus 1 end fraction..... left parenthesis straight i right parenthesis
1 equals straight A open parentheses straight t squared plus 1 close parentheses plus straight B open parentheses straight t squared plus straight t plus 1 close parentheses
Put space straight t equals negative 1 space and space straight t equals 0 space we space get space two space equations space straight A plus straight B equals 1 space and space 2 straight A plus straight B equals negative 1
Solving space these space we space get space straight A equals negative 2 space and space straight B equals 3. space Put space it space in space left parenthesis straight i right parenthesis space and space integrate space it space further. end style

Answered by Sneha shidid | 28th Jan, 2019, 09:13: AM