Integrating $\int_0^\infty \frac{\ln t}{t^2+a^2}\,dt$ with residue theorem.

Question

I want to calculate

$$\int_0^\infty \frac{\ln t}{t^2+a^2}\,dt$$

using the Residue Theorem. The contour I want to use is almost the upper half circle of radius $R$, but going around $0$ following a half-circle of radius $\epsilon$, and using the branch cut at $7\pi/8$. The picture is below.