If `f(x)` is continuous for all real values of `x ,` then `sum_(r=1)^nf(r-1+x)dxi se q u a lto` `int_0^nf(x)dx` (b) `int_0^1f(x)dx` `nint_0^1f(x)dx` (d) `(n-1)int_0^1f(x)dx`

Options

A. `int_(0)^(n)f(x)dx`
B. `int_(0)^(1)f(x)dx`
C. `nint_(0)^(1)f(x)dx`
D. `(n-1)int_(0)^(1)f(x)dx`

Answer

A

Explanation

`sum_(r=1)^(n)int_(0)^(1)f(r-1+x)dx`
`=int_(0)^(1)f(x)dx+int_(0)^(1)f(1+x)dx+int_(0)^(1)f(2+x)dx+`……………
`+ int_(0)^(1) f(n-1+x)dx`
`= int_(0)^(1) f(x) dx+int_(1)^(2)f(x)dx+int_(2)^(3) f(x)dx+int(r-1)^(2)f(x)dx+`………………
`+int_(n-1)^(n)f(x)dx`.
`=int_(0)^(n)f(x)dx`.