Question

$\int \frac{4}{x+logx}dx =$

Answer 1

4log|x+log x|+C

Answer 2

The integral as stated, $\int \frac{4}{x+\log x}dx$, is a non-elementary integral and cannot be solved using standard elementary functions. However, in the context of typical competitive exams like JEE/NEET, such questions usually imply a subtle typo that makes the integral solvable. The most common pattern for integrals involving a sum in the denominator is if the numerator is the derivative of the denominator (or a multiple thereof).

If we assume the numerator was intended to be $4$ times the derivative of the denominator, i.e., $4(1+\frac{1}{x})$, then a simple substitution works.

Let $u = x+\log x$.
Then $du = (1+\frac{1}{x})dx$.
The integral transforms to $\int \frac{4}{u}du$, which integrates to $4\log|u|+C$.
Substituting back $u = x+\log x$, we get $4\log|x+\log x|+C$.