How do you integrate # (x+1)/( (x^2 )*(x-1) )# using partial fractions?

Question

Cameron Alexander · Accepted Answer

See answer below:

Continuation:

William Abdalla · Answer

undefined To integrate $ \frac{x+1}{x^2(x-1)} $ using partial fractions, follow these steps:

1. Express the fraction as a sum of simpler fractions using partial fraction decomposition.
2. Determine the unknown constants.
3. Integrate each term separately.

Given fraction: $ \frac{x+1}{x^2(x-1)} $

Step 1: Partial fraction decomposition:

$$ \frac{x+1}{x^2(x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1} $$

Step 2: Determine the unknown constants $ A, B, $ and $ C $:

Multiply both sides by the denominator to clear fractions:

$$ x + 1 = A(x-1) + Bx(x-1) + Cx^2 $$

Expand and group like terms:

$$ x + 1 = Ax - A + Bx^2 - Bx + Cx^2 $$

$$ x + 1 = (B + C)x^2 + (A - B)x - A $$

Compare coefficients of corresponding powers of $ x $:

For $ x^2 $: $ B + C = 0 $  
For $ x $: $ A - B = 1 $  
For the constant term: $ -A = 1 $

Solve this system of equations to find the values of $ A, B, $ and $ C $.

Step 3: Integrate each term separately once you've found $ A, B, $ and $ C $.

$$ \int \frac{x+1}{x^2(x-1)} \, dx = \int \left( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1} \right) \, dx $$

Integrate each term separately:

$$ \int \frac{A}{x} \, dx + \int \frac{B}{x^2} \, dx + \int \frac{C}{x-1} \, dx $$

$$ = A\ln|x| - \frac{B}{x} + C\ln|x-1| + D $$

Where $ D $ is the constant of integration.