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for x lying between 1 to 2 x^2 lies between 1 and 4
for x lying between 1 to sqrt(2) x^ 2 lies between 1 and 2 whose [ ] gives 1 so putting 1 integrate.
then for x lying between sqrt(2) x^ 2 to sqrt(3) lies between 2 and 3 whose [ ] gives 2 so putting 2 integrate.
then for x lying between sqrt(3) x^ 2 to 2 lies between 3 and 4 whose [ ] gives 3 so putting 3 integrate.
in these kind of questions all you need is to break the interval in such a way that you get known values of the function.
I hope you will find my explanation helpful.
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