\( \int _ { \theta } ^ { \infty } \theta t ^ { - 1 } d t \)

To evaluate the integral \( \int _ { \theta } ^ { \infty } \theta t ^ { - 1 } d t \), we can factor out the constant \( \theta \) from the integral. This gives us: \[ \theta \int _ { \theta } ^ { \infty } t ^ { - 1 } d t \] The integral \( \int t^{-1} dt \) is the natural logarithm, so we need to compute: \[ \int _ { \theta } ^ { \infty } \frac{1}{t} dt = \lim_{b \to \infty} \int _ { \theta } ^ { b } \frac{1}{t} dt = \lim_{b \to \infty} \left[ \ln |t| \right] _ { \theta } ^ { b } \] This becomes: \[ \lim_{b \to \infty} (\ln b - \ln \theta) = \infty \] Therefore, the original integral diverges to infinity, and we have: \[ \int _ { \theta } ^ { \infty } \theta t ^ { - 1 } d t = \infty \]