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Evaluate: ∫(logx)2 dx - Mathematics and Statistics

Sum

Evaluate: `int (log "x")^2` dx

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Solution

Let I = `int (log "x")^2` dx

`= int (log "x")^2 * 1`dx

`= (log "x")^2 int 1 * "dx" - int ["d"/"dx" (log "x")^2 int 1 * "dx"]`dx

`= "x"(log "x")^2 * "x" - int 2 log "x" * 1/"x" * "x" * "dx"`

`= "x"(log "x")^2 - 2 int (log "x") * 1 * "dx"`

`= "x"(log "x")^2 - 2[log "x" int 1 * "dx" - int {"d"/"dx" (log "x") int 1 * "dx"}]`dx

`= "x"(log "x")^2 - 2[(log "x")"x" - int 1/"x" * "x" * "dx"]`

`= "x"(log "x")^2 - 2["x" log "x" - int 1 * "dx"]`

`= "x"(log "x")^2 - 2("x" log "x" - "x")` + c

∴ I = x(log x)2 - 2x log x - 2x + c

  Is there an error in this question or solution?
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 5 Integration
Miscellaneous Exercise 5 | Q 4.4 | Page 139
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