Final answer:
To find a general formula for the integral of f(x)/(x-c) from a to b, we use the Cauchy principal value if c is between a and b, or treat it as a regular improper integral otherwise. The calculation involves limits when c is within the bounds of integration. No universal formula exists without knowing specifics about f(x) and [a, b].
Step-by-step explanation:
To write a general formula for the integral \( \int_{a}^{b} \frac{f(x)}{x-c} dx \), where f(x) is an analytic function on the real line and c is a constant, we can use the Cauchy principal value if c is between a and b.
If c is not between a and b, the integral can be treated as a regular improper integral. For instances where c is within the bounds of integration, the integral is typically computed as the limit of two integrals as the point of singularity is approached from either side. This is usually expressed as:
\( P.V. \int_{a}^{b} \frac{f(x)}{x-c} dx = \lim_{\epsilon \to 0} \left( \int_{a}^{c-\epsilon} \frac{f(x)}{x-c} dx + \int_{c+\epsilon}^{b} \frac{f(x)}{x-c} dx \right) \)
However, there is no 'one-size-fits-all' formula for integrals of this type, as the expression and method depend on the specifics of the function f(x) and the interval [a, b]. Integral tables or computational tools like symbolic algebra systems could be utilized to find a more explicit formula for specific cases of f(x).