Question

Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`

Answer 1

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