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  4. If for every integer n,∫nn+1f(x) dx=n2, then the value of ∫-24f(x) dx is

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Q. If for every integer $ n,\int_{n}^{n+1}{f(x)}\,dx={{n}^{2}}, $ then the value of $ \int_{-2}^{4}{f(x)}\,dx $ is

Rajasthan PETRajasthan PET 2012

Solution:

Given, $ \int_{n}^{n+1}{f(x)dx}={{n}^{2}} $
On putting $ n=-2,-1,0,1,2,3, $ we get
$ \int_{-2}^{-1}{f(x)}\,dx=4\int_{-1}^{0}{f(x)}\,dx=1 $
$ \int_{0}^{1}{f(x)}\,dx=0\int_{1}^{2}{f(x)}\,dx=1 $
$ \int_{2}^{3}{f(x)}\,dx=4\int_{3}^{4}{f(x)}\,dx=9 $
$ \therefore $ $ \int_{-2}^{4}{f(x)}\,dx=4+1+0+1+4+9=19 $