perform the partial fraction decomposition of integrate (2x ^ 2 - 1)/((4x - 1)(x ^ 2 + 1)) dx

To perform the partial fraction decomposition of the integral (2x ^ 2 - 1)/((4x - 1)(x ^ 2 + 1)) dx, follow these steps:

Step 1: Set up the partial fraction decomposition. The given integral has a quadratic term in the denominator, so we set it up as follows:

(2x^2 - 1)/((4x - 1)(x^2 + 1)) = A/(4x - 1) + (Bx + C)/(x^2 + 1)

Step 2: Clear the fraction by multiplying through by the common denominator:

2x^2 - 1 = A(x^2 + 1) + (Bx + C)(4x - 1)

Step 3: Expand and collect like terms:

2x^2 - 1 = Ax^2 + A + 4Bx^2 - Bx + 4Cx - C

Step 4: Equate coefficients for the powers of x on both sides of the equation. This gives us a system of linear equations:

For x^2: 2 = A + 4B For x: 0 = -B + 4C For constants: -1 = A - C

Step 5: Solve this system of equations to find the values of A, B, and C.

Step 6: Substitute these values back into the partial fraction decomposition:

∫(2x^2 - 1)/((4x - 1)(x^2 + 1)) dx = ∫A/(4x - 1) dx + ∫(Bx + C)/(x^2 + 1) dx

Step 7: Now, you can integrate each term separately. The integral of A/(4x - 1) is straightforward. For the integral of (Bx + C)/(x^2 + 1), you can use a u-substitution (let u = x^2 + 1) or recognize it as a standard form of an integral that results in an arctan function.

Remember to add the constant of integration at the end.