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

Question

If for every integer n,∫nn+1f(x) dx=n2, then the value of ∫-24f(x) dx is (A) 16 (B) 14 (C) 19 (D) None of these

Tardigrade Admin · Accepted Answer

Given, ∫nn+1f(x)dx=n2 On putting n=-2,-1,0,1,2,3, we get ∫-2-1f(x) dx=4∫-10f(x) dx=1 ∫01f(x) dx=0∫12f(x) dx=1 ∫23f(x) dx=4∫34f(x) dx=9 ∴ ∫-24f(x) dx=4+1+0+1+4+9=19

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

16

14

19

None of these

Given, ∫nn+1​f(x)dx=n2 On putting n=−2,−1,0,1,2,3, we get ∫−2−1​f(x)dx=4∫−10​f(x)dx=1 ∫01​f(x)dx=0∫12​f(x)dx=1 ∫23​f(x)dx=4∫34​f(x)dx=9 ∴ ∫−24​f(x)dx=4+1+0+1+4+9=19

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