InterviewSolution
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InterviewSolution
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`int_(-1)^(2)|x^(3)-x|dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int_(-1)^(2)|x^(3)-x|dx` तथा `f(x)=|x^(3)-x|=|x(x^(2)-1)|=|x||x-1||x+1|` `={{:(-x xx-(x-1)xx(x+1)",","यदि ",-1lt xle0),(x xx(1-x)xx(x+1)",","यदि ",0lt x le1),(x(x-1)(x+1)",","यदि ",1ltxle2):}` `={{:(x^(3)-x",","यदि ",-1lt x le0),(x-x^(3)",","यदि ",0lt x le1),(x^(3)-x",","यदि ",1ltxle2):}` `therefore" "I=int_(-1)^(0)f(x)dx+int_(0)^(1)f(x)dx+int_(1)^(2)f(x)dx` `=int_(-1)^(0)(x^(3)-x)dx+int_(0)^(1)(x-x^(3))dx+int_(1)^(2)(x^(3)-x)dx` `=[(x^(4))/(4)-(x^(2))/(2)]_(1)^(0)+[(x^(2))/(2)-(x^(4))/(4)]_(0)^(1)+[(x^(4))/(4)-(x^(2))/(2)]_(1)^(2)` `=[0-0-{(1)/(4)-(1)/(2)}]+[(1)/(2)-(1)/(4)-0]+[(16)/(4)-(4)/(2)-{(1)/(4)-(1)/(2)}]` `=[(1)/(4)]+[(1)/(4)]+[4-2-(-(1)/(4))]` `=(1)/(4)+(1)/(4)+2+(1)/(4)=(11)/(4)` |
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