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Euler's substitution: Integrals of the form intR(x, sqrt(ax^(2)+bx+c))dx are claculated with the aid of one of the followingthree Euler substitutions: i.sqrt(ax^(2)+bx+c)=t+-x sqrt(a)if a gt 0 ii.sqrt(ax^(2)+bx+c)=tx+-x sqrt(c)if c gt 0 iii.sqrt(ax^(2)+bx+c)=(x-a)t if ax^(2)+bx+c=a(x-a)(x-b) i.e., if alpha is real root ofax^(2)+bx+c=0 Which of the following functions does not apear in the primitive of (dx)/(x+sqrt(x^(2)-x+1)) if t is a function of x? |
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Answer» `log_(e)|t|` SINCE here `c=1 gt 0, ` we can APPLY the SECOND Euler substitution: `sqrt(x^(2)-x+1)=tx-1` ` or (2t-1)x=(t^(2)-1)x^(2),x=(2t-1)/(t^(2)-1)` Substitutinginto I, we GET an integral of a RATIONAL fraction: `int (dx)/(x+sqrt(x^(2)-x+1))=int(-2t^(2)+2t-2)/(t(t-1)(t+1)^(2))dt` Now, `(-2t^(2)+2t-2)/(t(t-1)(t+1))=(A)/(t)+(B)/(t-1)+(D)/((t+1)^(2))+(E)/(t+1)` |
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