Indefinite integral are written in the form of `int f(x) dx = F(x) +C`

where: `f(x)` as the integrand

`F(x)` as the anti-derivative function

`C` as the arbitrary constant known as constant of integration

To evaluate the given integral problem: `int 1/(t(1+(ln(t))^2)) dt` , we may apply u-substitution by letting: `u =ln(t)` then `du = 1/t dt` .

Plug-in `u =ln(t)` and `du= 1/tdt` on `int 1/(t(1+(ln(t))^2)) dt` , the integral becomes:

`int 1/(t(1+(ln(t))^2)) dt =int1/(1+(ln(t))^2) *1/t dt`

`=int1/(1+u^2) du`

From the integration table, we have indefinite integration formula for rational function as: `int 1/(1+x^2) dx=arctan(x) +C` . The `int 1/(1+x^2)dx ` resembles the format of `int 1/(1+u^2) du ` where " u" corresponds to "x" .

This is our clue that we may apply the aforementioned formula for rational function.

We get: `int1/(1+u^2) du= arctan(u) +C` .

Plug-in `u =ln(t)` on `arctan(u) +C` , we get the indefinite integral as:

`int 1/(t(1+(ln(t))^2)) dt= arctan(ln(t))+C`