Final answer:
To integrate (x-1)/((x²)-4x+5), complete the square for the quadratic in the denominator and adjust the numerator to match the derivative of the completed square, resulting in the sum of ln and arctan functions plus a constant of integration.
Step-by-step explanation:
To find the integral of the function (x-1)/((x²)-4x+5), we first need to recognize that the denominator can't be factored into real linear factors due to the discriminant being negative ((-4)² - 4*1*5 < 0). In such cases, we often complete the square for the quadratic in the denominator to make the integration process more straightforward. The completion of the square for x² - 4x + 5 results in (x - 2)² + 1, where (x - 2) is the 'completed' square term and the constant 1 ensures we have an equivalent expression to the original denominator.
The next step is to write the numerator (x - 1) in a form that relates to the derivative of the denominator. This involves splitting the numerator or adding and subtracting a term so that part of the numerator becomes the derivative of (x - 2). For example, we can write (x - 1) as (x - 2) + 1. The integral then becomes the sum of two integrals: The integral of (x - 2)/((x - 2)² + 1) and the integral of 1/((x - 2)² + 1). The first part is a simple ln function due to the direct derivative form, and the second part can be solved using the arctan function since it has the standard form for the inverse tangent derivation.
The integral of the first part is ln|((x - 2)² + 1)| and the second part is arctan(x - 2). So, the entire integral would then be ln|((x - 2)² + 1)| + arctan(x - 2) + C, where C represents the constant of integration.