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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 followingfunctions does not appear in the primitive of(1)/(1+sqrt(x^(2)+2x+2)) if t is a function of x ? |
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Answer» `log_(e)|t+1|` `2x+2tx=t^(2)-2 or x=(t^(2)-2)/(2(1+t)) or dx=(t^(2)+2t+2)/(2(1+t)^(2))dt` `1+sqrt(x^(2)+2x+2)=1+t-(t^(2)-2)/(2(1+t))=(t^(2)+4t+4)/(2(1+t)).` Substitutinginto the integral, we get `I=int(2(1+t)(t^(2)+2t+2))/((t^(2)+4t+4)2(1+t)^(2))dt=int((t^(2)+2t+2)dt)/((1+t)(t+2)^(2))` Now, let us EXPAND the obtained proper rational fraction into partial fractions: `(t^(2)+2t+2)/((t+1)(t+2)^(2))=(A)/(t+1)+(B)/(t+2)+(D)/((t+2)^(2)).` |
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