1.

`f(x)= int_0^1 |t-x|t dt`

Answer» `f(x) = int_0^1|t-x|t dt `
When, `x le t`, then,
`f(x) = int_0^1 (t-x)t dt = int_0^1 (t^2-xt) dt`
`f(x) = [t^3/3-(xt^2)/2]_0^1 = [(1/3-x/2) -0 +0] = (2-3x)/6`
When, `x gt t`, then,
`f(x) = int_0^1 (x-t)t dt = int_0^1 (xt-t^2) dt`
`f(x) = [(xt^2)/2-t^3/3]_0^1 = [(x/2-1/3) -0 +0] = (3x-2)/6`


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