Use the partial ractions to find

The integral (3x^2-x+2)/(x-1)(x^2+1) dx

the integrand is x/(x^2+1)+2/(x-1)

so integral is

1/2 log (x^2+1) + 2 log(x-1)
= log [√(x^2+1)/(x-1)^2]

oops

log [√(x^2+1)*(x-1)^2]

To find the integral of a rational function like (3x^2 - x + 2) / [(x - 1)(x^2 + 1)], we can use the method of partial fractions. Here are the steps to solve it:

Step 1: Factor the denominator
The denominator is (x - 1)(x^2 + 1). Since x - 1 is already factored, the only remaining factor is x^2 + 1, which cannot be factored further because it is a sum of squares.

Step 2: Write the partial fraction form
We can write the rational function in the form of partial fractions as:

(3x^2 - x + 2) / [(x - 1)(x^2 + 1)] = A / (x - 1) + (Bx + C) / (x^2 + 1)

Step 3: Clear the fractions
To clear the fractions, we need to find the values of A, B, and C. We do this by equating the numerators:

(3x^2 - x + 2) = A(x^2 + 1) + (Bx + C)(x - 1)

Expanding the right side of the equation gives:

3x^2 - x + 2 = Ax^2 + A + Bx^2 - Bx + Cx - C

Now, we can compare the coefficients on both sides:

For x^2 terms: 3 = A + B
For x terms: -1 = -B + C
For constant terms: 2 = A - C

Step 4: Solve the system of equations
We have three equations with three unknowns (A, B, C). We can solve this system of equations to find the values of A, B, and C.

Solving the first and second equations simultaneously gives: A = 2, B = 1, C = 1.

Step 5: Rewrite the integral
Now that we have found the values of A, B, and C, we can rewrite the original integral in terms of the partial fractions:

∫(3x^2 - x + 2)/[(x - 1)(x^2 + 1)]dx = ∫(2/(x - 1) + (x + 1)/(x^2 + 1))dx

Step 6: Evaluate the integrals
The integral of 2/(x - 1) can be evaluated using the natural logarithm function:
∫(2/(x - 1))dx = 2ln|x - 1|

The integral of (x + 1)/(x^2 + 1) can be evaluated using arctan function:
∫((x + 1)/(x^2 + 1))dx = 1/2 * ln|x^2 + 1|

Putting it all together, the final solution is:

∫(3x^2 - x + 2)/[(x - 1)(x^2 + 1)]dx = 2ln|(x - 1)| + 1/2 * ln|(x^2 + 1)| + C

where C is the constant of integration.