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प्रश्न
Evaluate `int_(-1)^2|x^3-x|dx`
उत्तर
Let:
`I=int_(-1)^2|x^3-x|dx`
f(x)=x3−x
f(x)=x3−x=x(x−1)(x+1)
The signs of f(x) for the different values are shown in the figure given below:
f(x)>0 for all x∈(−1,0)∪(1,2)
f(x)<0 for all x∈(0,1)
Therefore
`|x^3-x|={(x^3-x,","xepsilon"(-1,0)"UU"(1,2)"),(-(x^3-x),","xepsilon(0,1)):}`
`:.I=int_(-1)^2|x^3-x|dx`
`=int_(-1)^0|x^3-x|dx+int_0^1|x^3-x|dx+int_1^2|x^3-x|dx`
`=int_(-1)^0(x^3-x)dx-int_0^1(x^3-x)dx+int_1^2(x^3-x)dx`
`=[x^4/4-x^2/2]_(-1)^0+[x^4/4-x^2/2]_0^1+[x^4/4-x^2/2]_1^2`
`=-(1/4-1/2)-(1/4-1/2)+(16/4-4/4)-(1/4-1/2)`
`=3/4+(4-2)`
`=11/4`
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