Find \(\int \frac{1}{s^{2}-2 s+5} \mathrm{~d} s\).

Partial fraction decomposition is a technique used to simplify complex rational expressions, especially when dealing with integrals. This method transforms a complicated fraction into a sum of simpler fractions, making integration possible. In the context of indefinite integrals, partial fraction decomposition allows for the integration of rational functions where the numerator has a lower degree than the denominator.

Typically, the process involves finding constants that satisfy the equation when the original fraction equals the sum of its decomposed parts. However, before applying this technique, we must first ensure that the denominator is factorized completely. If there are irreducible quadratic factors, like in our exercise, completing the square may be necessary to facilitate the decomposition or to rewrite the integral into a form that is easier to manage.

Completing the square is a powerful algebraic tool used to convert a quadratic polynomial into a perfect square trinomial, adding clarity to the solutions of quadratic equations and integrals. This method involves altering the structure of a quadratic equation so it can be expressed in the form \( (x-h)^2 + k \), making it simpler to analyze or integrate.

For the exercise at hand, the given denominator \( s^2-2s+5 \) is not an obvious perfect square. By completing the square, we rewrite it as \( (s-1)^2+4 \) which conveniently matches the format required to use the arctan integral formula. This technique is pivotal in manipulating the integral into a form where known integration strategies, such as the arctan formula, become applicable, thereby streamlining the solution process.

The arctan integral formula is a special integration rule that is utilized when encountering a function of the form \( \frac{1}{x^2+a^2} \). The formula is given by \( \int \frac{1}{x^2+a^2} \mathrm{~d} x = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C \), where \(+C\) represents the constant of integration.

In the exercise, after completing the square, we obtained an integral that fits perfectly within the parameters where the arctan integral formula can be applied. Recognizing such patterns is crucial for solving integrals efficiently. By substituting \(a\text{ and }x\) according to the transformed integral, we effectively find the antiderivative, showcasing the arctan integral formula's role as a fundamental solution technique in integral calculus.