Question

The function \(f(x) = \int \frac{x+3}{x^2-9x+20}\,dx\), then \(f(x)\) is

• A. increases on R
• B. decreases on R - (4, 5)
• C. decreases on \((-\infty ,-3]\cup (4,5)\)
• D. increases on \((-3,\infty )\)

Hint:

The sign of \(f'(x)=\frac{x+3}{(x-4)(x-5)}\) decides where \(f\) increases or decreases.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
If \(f\) is an antiderivative of \(g\), then \(f'(x)=g(x)\), so monotonicity follows the sign of \(g\).

Step 2: Key Formula or Approach
\(f'(x)=\dfrac{x+3}{x^2-9x+20}=\dfrac{x+3}{(x-4)(x-5)}\). Critical points: \(-3,4,5\).

Step 3: Detailed Explanation
For \(x<-3\): numerator negative, denominator positive, so \(f'<0\).
For \(-3<x<4\): numerator positive, denominator positive, so \(f'>0\).
For \(4<x<5\): numerator positive, denominator negative, so \(f'<0\).
For \(x>5\): both positive, so \(f'>0\).
Hence \(f\) decreases on \((-\infty,-3]\cup(4,5)\).

Final Answer:
The function decreases on \((-\infty,-3]\cup(4,5)\), option (C). \[ \boxed{(-\infty,-3]\cup(4,5)\ \text{(C)}} \]