Question

`int_0^x{int_0^uf(t)dx}d ui se q u a lto` `int_0^x(x-u)f(u)d u` `int_0^x uf(x-u)d u` `xint_0^xf(u)d u` (d) `xint_0^x uf(u-x)d u`
A. `int_(0)^(x)(x-u)f(u)du`
B. `int_(0)^(x) uf(x-u)du`
C. `x int_(0)^(x)f(u)du`
D. `x int_(0)^(x)uf(u-x)du`

Answer 1

Correct Answer - A::B
L.H.S `=int_(0)^(x) {int_(0)^(u)f(t)dt}du`
Integrating by parts choose 1 as the second function. Then,
L.H.S`={uint_(0)^(u)f(t)dt}_(0)^(x)-int_(0)^(x)f(u)u du`
`=x int_(0)^(x)f(t)dt-int_(0)^(x)f(u)u du`
`=x int_(0)^(x)f(u)du-int_(0)^(x)f(u) udu-int_(0)^(x)f(u)(x-u)du`
`=R.H.S`.