Question

Evaluate the integral: \(\int \frac{2x}{x^2 - 5x + 4} \, dx\)

• A. \( \frac{8}{3}\log |x-4| - \frac{2}{3}\log |x-1| + C \)
• B. \( \frac{2}{3}\log |x-4| - \frac{8}{3}\log |x-1| + C \)
• C. \( \frac{8}{3}\log |x-1| - \frac{2}{3}\log |x-4| + C \)
• D. \( \frac{2}{3}\log |x-1| - \frac{8}{3}\log |x-4| + C \)

Hint:

For partial fractions of the form \( \frac{f(x)}{(x-a)(x-b)} \), use the "cover-up" method:
To find \( A \), cover \( (x-4) \) and put \( x=4 \) in the rest of the expression: \( \frac{2(4)}{4-1} = 8/3 \).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The integrand is a proper rational function. We should use partial fraction decomposition to integrate it.

Step 2: Key Formula or Approach:
Factor the denominator: \( x^2 - 5x + 4 = (x-4)(x-1) \).
Let \( \frac{2x}{(x-4)(x-1)} = \frac{A}{x-4} + \frac{B}{x-1} \).

Step 3: Detailed Explanation:
Finding constants \( A \) and \( B \):
\[ 2x = A(x-1) + B(x-4) \]
Put \( x = 4 \):
\[ 2(4) = A(4-1) \Rightarrow 8 = 3A \Rightarrow A = \frac{8}{3} \]
Put \( x = 1 \):
\[ 2(1) = B(1-4) \Rightarrow 2 = -3B \Rightarrow B = -\frac{2}{3} \]
Substituting back into the integral:
\[ I = \int \left( \frac{8/3}{x-4} - \frac{2/3}{x-1} \right) \, dx \]
Integrating term by term:
\[ I = \frac{8}{3} \log |x-4| - \frac{2}{3} \log |x-1| + C \]

Step 4: Final Answer:
The value of the integral is \( \frac{8}{3}\log |x-4| - \frac{2}{3}\log |x-1| + C \).