<s>[INST] Evaluate: $\int \frac{1}{(x+a)(x+b)}$ Evaluate:
$$\int \frac{1}{(x+a)(x+b)}$$
My attempt:
$$\int \frac{1}{(x+a)(x+b)} = \frac{A}{x+a} + \frac{B}{x+b}$$
$$1 = A(x+b) + B(x+a)$$
$$x = -b$$
$$1 = A(-b + b) + B(-b + a)$$
$$1 = B(-b + a)$$
$$x = -a$$
$$1 = A(-a + b) + B(-a + a)$$
$$1 = A(-a + b)$$
At this point I have no idea how to proceed. Can someone help me with this? Please.
 [/INST] You've bagged it. You have
$${1\over (x-a)(x-b)} = {1\over b -a}\left({1\over x + a} - {1\over x  + b }\right).$$
Integrate to obtain
$$\int{dx\over (x-a)(x-b)} = {1\over b -a}(\ln|x + a| - \ln|x + b|) + C$$
Don't become afraid of all of the constants running around.
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