InterviewSolution
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InterviewSolution
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Evaluate : (i) `int(3x^(2))/((1+x^(6)))dx` (ii) `int(x^(3))/((x^(2)+1)^(3))dx` (iii) `(x^(8))/((1-x^(3))^(1//3))dx` |
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Answer» (i) Put `x^(3)=t` so that `3x^(2)dx=dt`. `:.int(3x^(2))/((1+x^(6)))dx=int(dt)/((1+t^(2)))=tan^(-1)t+C=tan^(-1)x^(3)+C`. (ii) Put `(x^(2)+1)` =t so that `x^(2)=(t-1)andxdx=(1)/(2)dt`. `:.int(x^(3))/((x^(2)+1)^(3))dx=int(x^(2)*x)/((x^(2)+1))dx` `=(1)/(2)int((t-1))/(t^(3))dt=(1)/(2)int(1)/(t^(2))dt-(1)/(t^(3))dt` `=(-1)/(2t)+(1)/(4t^(2))+C=(-1)/(2(x^(2)+1))+(1)/(4(x^(2)+1)^(2))+C` `=(-1+2x^(2))/(4(x^(2)+1)^(2))+C`. (iii) Put `(1=x^(3))=t` so that `x^(3)=(1-t)andx^(2)dx=-(1)/(3)dt`. `:.int(x^(8))/((1-x^(3))^(1//3))dx=int(x^(6)*x^(2))/((1-x^(3))^(1//3))dx` `=-(1)/(3)int((1-t)^(2))/(t^(1//3))dt=-(1)/(3)((1+t^(2)-2t))/(t^(t//3))dt` `=-(1)/(3)intt^(-1//3)dt-(1)/(3)intt^(5//3)dt+(2)/(3)intt^(2//3)dt` `=-(1)/(2)t^(2//3)-(1)/(8)t^(8//3)+(2)/(5)t^(5//3)+C` `=-(1)/(2)(1-x^(3))^(2//3)-(1)/(8)(1-x^(3))^(8//3)+(2)/(5)(1-x^(3))^(5//3)+C`. |
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