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Question
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
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Solution
Let I = `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Put log x = t
∴ `1/"x" "dx" = "dt"`
∴ I = `int "dt"/("t"^2 + 4"t" - 1)`
`= int 1/("t"^2 + 4"t" + 4 - 4 - 1)`dt
`= int 1/(("t + 2")^2 - 5)` dt
`= int 1/(("t + 2")^2 - (sqrt5)^2)` dt
`= 1/(2 sqrt5) log |("t" + 2 - sqrt5)/("t" + 2 + sqrt5)|` + c
∴ I = `1/(2 sqrt5) log|(log"x" + 2 - sqrt5)/(log "x" + 2 + sqrt5)|` + c
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