integrate (x^2 + 2x + 5) / (x - 2)
To integrate the given rational function, we can use the method of partial fractions. Here's how you can do it step by step:
Step 1: Factorize the denominator.
The denominator x - 2 cannot be factored any further since it is already in its simplest form.
Step 2: Write the partial fractions.
Since the degree of the numerator (2) is greater than the degree of the denominator (1), we start by performing polynomial long division to divide (x^2 + 2x + 5) by (x - 2). The result is (x + 4) with a remainder of 13.
So, the partial fraction decomposition can be written as:
(x^2 + 2x + 5) / (x - 2) = (x + 4) + 13 / (x - 2)
Step 3: Integrate the partial fractions.
Integrating the partial fractions separately:
∫ [(x + 4) + 13 / (x - 2)] dx = ∫ (x + 4) dx + ∫ (13 / (x - 2)) dx
The integral of (x + 4) with respect to x is (1/2)x^2 + 4x + C1, where C1 is the constant of integration.
For the integral of (13 / (x - 2)), we can use the natural logarithm function. The integral will be 13 ln|x - 2| + C2, where ln is the natural logarithm and C2 is the constant of integration.
Therefore, the final result after integrating the partial fractions is:
∫ [(x^2 + 2x + 5) / (x - 2)] dx = (1/2)x^2 + 4x + 13 ln|x - 2| + C, where C is the constant of integration.