# What is the definite integral of x*ln x between x = 4 and x = 10

The definite integral `int_4^10 x*ln x dx` has to be determined.

The integral can be determined using integration by parts. The integral `int u dv = u*v - int v du`

`int x*ln x dx`

let `u = ln x` , `dv = x dx`

=> `du = (1/x) dx` , `v = x^2/2`

`int x*ln x dx`

= `ln x*x^2/2 - int x^2/2*(1/x) dx`

= `ln x*x^2/2 - int x/2 dx`

= `(x^2*ln x)/2 - x^2/4 + C`

`int_4^10 x*ln x dx`

= `[(x^2*ln x)/2 - x^2/4 + C]_4^10`

= `(10^2*ln 10)/2 - (4^2*ln 4)/2 - 100/4 + 16/4`

= `50*ln 10 - 8*ln 4 - 21`

The definite integral `int_4^10 x*ln x dx = 50*ln 10 - 8*ln 4 - 21`

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