Fill in the Blank. ∫1x3[logxx]2dx=P(logx)3 + c, then P = _______ - Mathematics and Statistics

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Fill in the Blanks

Fill in the Blank.

`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______

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Solution

`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = `underline(1/3)`

Explanation:

Let I = `int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" * (log "x")^3` + c

I = `int 1/"x"^3 [log "x"^"x"]^2 "dx" = int 1/"x"^3 ("x log x")^2 * "dx"`

`=int 1/"x"^3 * "x"^2 * (log "x")^2 "dx" = int 1/"x" (log "x")^2 * "dx"`

∴ Put log x = t

∴ `1/"x"` dx = dt

∴ I = `int "t"^2 * "dt"`

`= "t"^3/3 + "c"`

`= 1/3 (log "x")^3 + "c"`

∴ P = `1/3`

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Chapter 5: Integration - MISCELLANEOUS EXERCISE - 5 [Page 138]

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Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 5 Integration
MISCELLANEOUS EXERCISE - 5 | Q II. 5. | Page 138

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