Antiderivative of the secant function

Compute the indefinite integral $\displaystyle\int\sec\theta\,d\theta$.

$\ln|\sec\theta+\tan\theta|+C$. Multiply top and bottom by $\sec\theta+\tan\theta$: the integrand becomes $\frac{\sec^2\theta+\sec\theta\tan\theta}{\sec\theta+\tan\theta}$, whose numerator is exactly the derivative of the denominator.