Logarithmic Integrals of Reciprocal Linear Functions over Bounded and Unbounded Intervals

Let $\log$ be the natural logarithm, $(a,s)$ the open interval and $(a,\infty)$ the open ray, and let integrability on a Borel set and $\int_{B}f(t)\,dt$ be as in Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §integral.

1. (Bounded intervals)¶ Let $a,s,b$ be real with $a<s$ and $a+b>0$. Then $t\mapsto1/(b+t)$ is integrable on $(a,s)$, and

$$\int_{(a,s)}\frac{dt}{b+t}=\log(b+s)-\log(b+a).$$

2. (Half-lines)¶ Let $a,b,c$ be real with $a+b>0$ and $a+c>0$. Then $t\mapsto\frac{1}{b+t}-\frac{1}{c+t}$ is integrable on $(a,\infty)$, and

$$\int_{(a,\infty)}\Bigl(\frac{1}{b+t}-\frac{1}{c+t}\Bigr)dt=\log(a+c)-\log(a+b).$$