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Question
Evaluate: ∫ (log x)2 dx
Solution
Let I = ∫ (log x)2 dx
I = ∫ (log x)2 . 1 dx
I = `(log x)^2 int 1. "dx" − int ["d"/"dx" (log x)^2 int 1. "dx"] "dx"`
I = `x(log x)^2 − int 2 log x. 1/cancelx. cancelx "dx"`
I = `x(log x)^2 − 2 int log x. 1 "dx"`
I = `x(log x)^2 − 2[log x int 1. "dx" − int {"d"/"dx" (log x) int 1. "dx"}]`dx
I = `x(log x)^2 − 2[(log x)x − int 1/cancelx. cancelx. "dx"]`
I = `x(log x)^2 − 2[xlog x − int 1. "dx"]`
I = `x(log x)^2 − 2(x log x − x) + c`
∴ I = x(log x)2 − 2x log x + 2x + c
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