Calculus of a Single Variable, Chapter 8, 8.6, Section 8.6, Problem 26

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