Final answer:
To evaluate the indefinite integral, we can decompose the fraction into partial fractions. After equating coefficients, we can integrate each term separately.
Step-by-step explanation:
To evaluate the indefinite integral, we need to decompose the fraction into partial fractions. The denominator factorizes into (x + 2)(x - 1)(x - 1). Therefore, the partial fraction decomposition is:
(5x + 1)/((x + 2)(x - 1)(x - 1)) = A/(x + 2) + B/(x - 1) + C/(x - 1)
Multiplying through by the common denominator, we get:
5x + 1 = A(x - 1)(x - 1) + B(x + 2)(x - 1) + C(x + 2)
Expanding and equating coefficients, we find that A = 2, B = 1, and C = -1.
Now, we can integrate each term separately:
∫ A/(x + 2) dx = 2ln|x + 2| + K1
∫ B/(x - 1) dx = ln|x - 1| + K2
∫ C/(x - 1) dx = -ln|x - 1| + K3
Therefore, the indefinite integral of (5x + 1)/((x + 2)(x - 1)(x - 1)) dx is 2ln|x + 2| + ln|x - 1| - ln|x - 1| + C, where C is the constant of integration.