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Question
Evaluate: ∫ x . log x dx
Solution
Let ∫ x . log x dx
= ∫ log x . x . dx ...[by LIATE rule]
Integrating by parts
I = `"log x" . int "x" "dx" - int ["d"/"dx" ("log x") int "x" "dx"] "dx"`
`= "log x" . ("x"^2/2) - int 1/"x" . ("x"^2/2) "dx"`
`= "x"^2/2 . "log x" - 1/2 int "x" "dx"`
`= "x"^2/2 . "log x" - 1/2 ("x"^2/2) + "c"`
`= "x"^2/2 . "log x" - "x"^2/4" + c`
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