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The total number for distinct x epsilon[...

The total number for distinct `x epsilon[0,1]` for which `int_(0)^(x)(t^(2))/(1+t^(4))dt=2x-1` is __________.

Text Solution

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The correct Answer is:
1
Let `f(x)=int_(0)^(x)(t^(2))/(1+t^(4))dt rArrf'(x)=(x^(2))/(1+x^(4))gt0," for all "x in [0,1]`
`therefore f(x)` is increasing.
At `x=0, f(0)=0 and " at " x=1`,
`f(1)=int_(0)^(1) (t^(2))/(1+t^(4))dt`
Because, `0 lt (t^(2))/(1+t^(4)) lt (1)/(2) rArr int_(0)^(1) 0*dt lt int_(0)^(1)(t^(2))/(1+t^(4))dt lt int_(0)^(1) (1)/(2) *dt`
`rArr 0 lt f(1) lt (1)/(2)`
Thus, f(x) can be plotted as

`therefore y=f(x) and y=2x-1` can be shown as

From the graph, the total number of distinct solutions for `x in (0,1] =1. " "`[as they intersect only at one point]
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