Question

Evaluate:

`∫dx/(x[(log x)^2 + 5logx + 6])`

Answer 1

`∫dx/(x[(log x)^2 + 5logx + 6])`

let log x = t

On differentiating 

`1/x = dt/dx`

⇒ dt = `dx/x`       ...[i]

⇒ `∫ dt/(t^2 + 5t + 6)`

On making perfect square       ...[From (i)]

⇒ `∫ dt/(t^2 + 2 xx t xx 5/2 + 25/4 − 25/4 + 6)`

⇒ `∫ dt/((t + 5/2)^2 − (1/2)^2)`

⇒ `1/(2 × 1/2)[log|(t + 5/2 − 1/2)/(t + 5/2 + 1/2)|] + c`     ...`[∵ ∫dx/(x^2 − a^2) ⇒ 1/(2a)log|(x − a)/(x + a)| + c]`

⇒ `log|(t + 2)/(t + 3)| + c`

⇒ `log|(logx + 2)/(logx + 3)| + c`