The integral `int_1^e x ln x dx`Could you  me a detailed explaination,thank for your  reconstellation in advance

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate definite integral `int_1^e x*ln x dx`  using parts such that:

`int udv = uv - int vdu`

You need to consider u = ln x and dv = xdx such that:

`u = ln x =gt du = (dx)/x`

`dv = xdx =gt v = (x^2)/2`

`int_1^e x*ln x dx = (x^2*ln x)/2|_1^e - int_1^e (x^2)/(2x)dx`

`int_1^e x*ln x dx = (x^2*ln x)/2|_1^e - (1/2)int_1^e xdx`

`int_1^e x*ln x dx = (x^2*ln x)/2|_1^e - (x^2)/4|_1^e`

`int_1^e x*ln x dx = (e^2*ln e - 1*ln1)/2 - (e^2 -1)/4`

You should substitute 1 for ln e and 0 for ln1 such that:

`int_1^e x*ln x dx = (e^2 -1)/4`

Hence, evaluating the definite integral yields `int_1^e x*ln x dx = (e^2 -1)/4.`


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