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Question
Integrate the function in x (log x)2.
Solution
Let `I = int x (log x)^2 dx`
`= int (log x)^2 * x dx`
`= (log x)^2 int x dx - int [d/dx (log x)^2 * int x dx] dx`
`= x^2/2 (log x)^2 - int (log x) * x dx + C`
`= x^2/2 (log x)^2 - [ (log x) * x^2/2 - int 1/x * x^2/2 dx]`
`= x^2/2 (log x)^2 - x^2/2 log x + 1/2 int x dx`
`= x^2/2 (log x)^2 - x^2/2 log x + 1/2 int*x^2/2 + C`
`= x^2 (log x)^2 - x^2/2 log x + 1/2 * x^2/2 + C`
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