Question

Solve: \( \int \frac{(3x + 5)dx}{x^3 - x^2 - x + 1} \)

Hint:

When performing partial fraction decomposition, using the "cover-up" method (substituting the roots of the denominator) is the fastest way to find coefficients for distinct linear factors and the coefficient of the highest power of a repeated linear factor. For the remaining coefficients, you can substitute any other convenient number (like x=0) or compare the coefficients of the highest power of x on both sides.

Answer 1

Step 1: Understanding the Concept:
This is an integral of a rational function. Since the degree of the numerator is less than the degree of the denominator, we can solve it using the method of partial fraction decomposition. This involves factoring the denominator and expressing the rational function as a sum of simpler fractions.
Step 2: Key Formula or Approach:
1. Factor the denominator polynomial.
2. Set up the partial fraction expansion based on the factors.
3. Solve for the unknown coefficients.
4. Integrate the resulting simpler fractions.
Step 3: Detailed Explanation or Calculation:
1. Factor the Denominator:
Let the denominator be \( D(x) = x^3 - x^2 - x + 1 \). We can factor it by grouping:
\[ D(x) = x^2(x - 1) - 1(x - 1) \] \[ D(x) = (x^2 - 1)(x - 1) \] \[ D(x) = (x - 1)(x + 1)(x - 1) = (x - 1)^2(x + 1) \] The denominator has a repeated linear factor \( (x-1) \) and a distinct linear factor \( (x+1) \).
2. Partial Fraction Decomposition:
We can write the integrand as:
\[ \frac{3x + 5}{(x - 1)^2(x + 1)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{C}{x + 1} \] To find the coefficients A, B, and C, multiply both sides by the denominator:
\[ 3x + 5 = A(x - 1)(x + 1) + B(x + 1) + C(x - 1)^2 \] Now, substitute strategic values for x:
- For \( x = 1 \): \( 3(1) + 5 = A(0) + B(1+1) + C(0) → 8 = 2B → B = 4 \).
- For \( x = -1 \): \( 3(-1) + 5 = A(0) + B(0) + C(-1-1)^2 → 2 = 4C → C = \frac{1}{2} \).
- For \( x = 0 \): \( 3(0) + 5 = A(-1)(1) + B(1) + C(-1)^2 → 5 = -A + B + C \).
Substitute the values of B and C:
\[ 5 = -A + 4 + \frac{1}{2} → 5 = -A + \frac{9}{2} → A = \frac{9}{2} - 5 = \frac{9-10}{2} = -\frac{1}{2} \] 3. Integrate:
Now, we can rewrite the integral as:
\[ \int \left( \frac{-1/2}{x - 1} + \frac{4}{(x - 1)^2} + \frac{1/2}{x + 1} \right) dx \] \[ = -\frac{1}{2}\int \frac{1}{x - 1} dx + 4\int \frac{1}{(x - 1)^2} dx + \frac{1}{2}\int \frac{1}{x + 1} dx \] \[ = -\frac{1}{2}\ln|x - 1| + 4\left(\frac{(x-1)^{-1}}{-1}\right) + \frac{1}{2}\ln|x + 1| + C \] \[ = -\frac{1}{2}\ln|x - 1| - \frac{4}{x - 1} + \frac{1}{2}\ln|x + 1| + C \] Step 4: Final Answer:
The solution is:
\[ \frac{1}{2}\ln|x + 1| - \frac{1}{2}\ln|x - 1| - \frac{4}{x - 1} + C \] This can also be written using logarithm properties as:
\[ \frac{1}{2}\ln\left|\frac{x + 1}{x - 1}\right| - \frac{4}{x - 1} + C \]