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Evaluate the following Integrals. int...

Evaluate the following Integrals.
`int x (log x)^(2) dx`

Text Solution

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`" Let " I= int x (log x)^(2) dx`
` I= (log x)^(2) int x dx - int [(d)/(dx) (log x)^(2) int x dx ]dx`
`=(log x)^(2) .(x^(2))/(2) - int ((2 log x)/(x).(x^(2))/(2))dx`
`= (x^(2))/(2) (log x)^(2) - int x log x dx`
`=(x^(2))/(2) (log x)^(2) -[ log x int x dx - int ((d)/(dx) log x) int x dx } dx]+c`
` =(x^(2))/(2) (log x)^(2) -((x^(2))/(2) log x-int (1)/(x).(x^(2))/(2) dx )+C`
` =(x^(2))/(2) (log x)^(2) -(x^(2))/(2) log + (1)/(2) int x dx+C`
` =(x^(2))/(2) (log x)^(2) -(x^(2))/(2) log + (x^(2))/(4)+C`
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