Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 17 / a / Solution

Use integration by substitution $t=\tan x$, so $dt=dx/{\cos }^{2}x$. The indefinite integral becomes $\int dt/[t(t^2-1)]$. Its partial fraction decomposition isIntegrating gives $-\ln |t|+\frac{1}{2}\ln |t^2-1|+C$. HenceThe formula applies on each interval where the printed integrand is defined: $\cos x\ne 0$ and $\tan x\notin \{0,1,-1\}$. The constant of integration may differ between such intervals.