Question

Gives f(x) with constant of integration zero then find f(1)

$\int \frac{\log x}{{(1+\log x)}^2}dx$

Answer 1

Substitute $\log x=t;\therefore x=e^t$ and $dx=e^{t}dt.$
$\therefore I=\int \frac{\log x}{{(1+\log x)}^2}dx$
$=\int \frac{t{e}^{t}dt}{{(1+t)}^2}=e^t.\frac{1}{t+1},$ by part (a).
$=\frac{x}{1+\log x}.$