Please solve 

Asked by mforeachother10 | 16th Jul, 2020, 02:33: PM

Expert Answer:

TO find the integral of   (x2-1)/(xsqrt{x2 + alpha x + 1} sqrt{x2 + beta x + 1})
 
straight I equals integral fraction numerator straight x squared minus 1 over denominator straight x square root of straight x squared plus αx plus 1 end root space square root of straight x squared plus βx plus 1 end root end fraction dx
equals integral fraction numerator straight x squared open parentheses 1 minus begin display style 1 over straight x squared end style close parentheses over denominator straight x squared space square root of straight x plus straight alpha plus begin display style 1 over straight x end style end root space square root of straight x plus straight beta plus begin display style 1 over straight x end style end root end fraction dx
equals integral fraction numerator open parentheses 1 minus begin display style 1 over straight x squared end style close parentheses over denominator square root of straight x plus straight alpha plus begin display style 1 over straight x end style end root space square root of straight x plus straight beta plus begin display style 1 over straight x end style end root end fraction dx
Take space straight x plus 1 over straight x equals straight t space space rightwards double arrow space open parentheses 1 minus 1 over straight x squared close parentheses dx equals dt
rightwards double arrow straight I equals integral fraction numerator dt over denominator square root of straight alpha plus straight t end root space square root of straight beta plus straight t end root end fraction equals integral fraction numerator dt over denominator square root of αβ plus open parentheses straight alpha plus straight beta close parentheses straight t plus straight t squared end root space end fraction equals integral fraction numerator dt over denominator square root of αβ plus open parentheses straight alpha plus straight beta close parentheses straight t plus straight t squared plus open parentheses begin display style fraction numerator straight alpha plus straight beta over denominator 2 end fraction end style close parentheses squared minus open parentheses begin display style fraction numerator straight alpha plus straight beta over denominator 2 end fraction end style close parentheses squared end root space end fraction
rightwards double arrow straight I equals integral fraction numerator dt over denominator square root of open parentheses begin display style straight t plus fraction numerator straight alpha plus straight beta over denominator 2 end fraction end style close parentheses squared minus open parentheses begin display style fraction numerator straight alpha minus straight beta over denominator 2 end fraction end style close parentheses squared end root space end fraction
rightwards double arrow straight I equals log open vertical bar open parentheses straight t plus fraction numerator straight alpha plus straight beta over denominator 2 end fraction close parentheses minus square root of open parentheses straight t plus straight alpha close parentheses open parentheses straight t plus straight beta close parentheses end root close vertical bar plus straight c

Answered by Renu Varma | 23rd Jul, 2020, 11:39: AM

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