InterviewSolution
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InterviewSolution
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Evaluate : (i) `int(dx)/((9x^(2)-1))` (ii) `int(x)/((x^(4)-9))dx` (iii) `int(x^(2))/((x^(2)-9))dx` |
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Answer» We have (i) `int(dx)/((9x^(2)-1))=(1)/(9)*int(dx)/((x^(2)-(1)/(9)))` `=(1)/(9)*int(dx)/({x^(2)-((1)/(3))^(2)})` `=(1)/(9)*(1)/((2xx(1)/(3)))log|(x-(1)/(3))/(x+(1)/(3))|+C [becauseint(dx)/((x^(2)-a^(2)))=(1)/(2a)log|(x-a)/(x+a)|+C]` `=(1)/(6)log|(3x-1)/(3x+1)|+C`. (ii) Putting `x^(2)=t " and "2xdx=dt` , we get `int(x)/((x^(4)-9))dx=(1)/(2)int(dt)/((t^(2)-9))=(1)/(2)*int(dt)/({t^(2)-(3)^(2)})` `=(1)/(2)*(1)/((2xx3))log|(t-3)/(t+3)|+C` `=(1)/(2)log|(x^(2)-3)/(x^(2)+3)|+C`. (iii) `int(x^(2))/((x^(2)-9))dx=int{1+(9)/(x^(2)-9)}dx` `=intdx+9int(dx)/({x^(2)-(3)^(2)})` `=x+9*[(1)/((2xx3))log|(x-3)/(x+3)|]+C` `=x+(3)/(2)log|(x-3)/(x+3)|+C`. |
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