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If int0^x {int0^u f(t) dx}du is equal t...

If `int_0^x {int_0^u f(t) dx}du` is equal to (a) `int_0^x (x-u) f(u)` (b) `int_0^xuf(x-u)du` (c) `x int_0^x f(u) du` (d) `x int_0^x uf(u-x)du`

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`

Text Solution

Verified by Experts

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`.
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