InterviewSolution
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InterviewSolution
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Statement -1 : If `I_(1)=int(e^(x))/(e^(4x)+e^(2x)+1)dx` and `I_(2)=int(e^(-x))/(e^(-4x)+e^(-2x)+1)dx`, then `I_(2)-I_(1)=(1)/(2)log((e^(2x)-e^(x)+1)/(e^(2x)+e^(x)+1))+C` where C is an arbitrary constant. Statement -2 : A primitive of f(x) `=(x^(2)-1)/(x^(4)+x^(2)+1)` is `(1)/(2)log((x^(2)-x+1)/(x^(2)+x+1))`.A. Statement - 1 True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.B. Statement - 1 is True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.C. Statement - 1 True ,Statement - 2 is False.D. Statement - 1 is False , Statement - 2 is True. |
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Answer» A primitive of f (x) `=(x^(2)-1)/(x^(4)+x^(2)+1)` is given by `I=int(x^(2)=1)/(x^(4)+x^(2)+1)dx=int(1-(1)/(x^(2)))/(x^(2)+(1)/(x^(2))+1)dx` `rArrI=int(1)/((x+(1)/(x))^(2)-1^(2))d(x+(1)/(x))=(1)/(2)log|(x+(1)/(x)-1)/(x+(1)/(x)+1)|+C` `rArrI=int(1)/(2)log((x^(2)-x+1)/(x^(2)+x+1))+C` So , statement - 2 is true. Now , `I_(2)-I_(1)=int(e^(-x))/(e^(-4x)+e^(-2x)+1)dx-int(e^(x))/(e^(4x)+e^(2x)+1)dx` `rArrI_(2)-I_(1)=int(e^(3x))/(e^(4x)+e^(2x)+1)dx-int(e^(x))/(e^(4x)+e^(2x)+1)dx` `rArrI_(2)-I_(1)=int(e^(3x))/(e^(4x)+e^(2x)+1)d(e^(x))` `rArrI_(2)-I_(1)=(1)/(2)log((e^(2x)-e^(x)+1)/(e^(2x)+e^(x)+1))+C` [ Using statment -2 ] So , statement - 1 is true . Also . statement - 2 is a correct explanation for statement -1. |
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