Evaluate ∫ (ln x)/((1 + ln x)²) , dx — JEE Mathematics

Let $t = \ln x \implies x = e^t \implies dx = e^t \, dt$.
Substitute into the integral:

$$I = \int \frac{t}{(1 + t)^2} e^t \, dt$$

Rewrite the numerator $t$ as $(1 + t) - 1$:

$$I = \int e^t \left[ \frac{1 + t}{(1 + t)^2} - \frac{1}{(1 + t)^2} \right] \, dt = \int e^t \left[ \frac{1}{1 + t} + \left(-\frac{1}{(1 + t)^2}\right) \right] \, dt$$

This matches the standard form $\int e^t [f(t) + f'(t)] \, dt = e^t f(t) + C$, where $f(t) = \frac{1}{1 + t}$.

$$I = \frac{e^t}{1 + t} + C$$

Substitute back $t = \ln x$ and $e^t = x$:

$$I = \frac{x}{1 + \ln x} + C$$

Answer: $\frac{x}{1 + \ln x} + C$