Calculus II · Integration techniques and improper integrals
Keep two divergent logarithms under one cutoff
Problem
For $a,b>0$, define \[ J(a,b)=\int_0^\infty\frac{dx}{(x+a)(x+b)}. \] First prove that the integral converges. For $a\ne b$, evaluate it by partial fractions, but keep both logarithmic terms under the same upper cutoff. Compute $J(a,a)$ directly. Finally prove that your formula for $a\ne b$ has limit $1/a$ as $b\to a$, and explain why writing it as a difference of two separately evaluated improper integrals is invalid.
Hint
For $a\ne b$, use $\dfrac1{(x+a)(x+b)}=\dfrac1{b-a}\left(\dfrac1{x+a}-\dfrac1{x+b}\right)$.
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