Question

\(\int _0^2|4x-5|\,dx = \ldots\)

• A. \(\frac{17}{4}\)
• B. \(\frac{18}{3}\)
• C. \(\frac{1}{25}\)
• D. \(\frac{13}{2}\)

Hint:

Split the integral at x = 5/4, where 4x-5 changes sign.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The absolute value changes the expression at the point where \(4x - 5 = 0\), that is \(x = \dfrac54\). This point lies inside \([0, 2]\), so we split the integral there.

Step 2: Key Formula or Approach:
\[ |4x-5| = \begin{cases}5 - 4x, & x < \tfrac54\\ 4x - 5, & x \ge \tfrac54\end{cases} \]

Step 3: Detailed Explanation:
First part:
\[ \int_0^{5/4}(5-4x)\,dx = \left[5x - 2x^2\right]_0^{5/4} = \frac{25}{4} - \frac{25}{8} = \frac{25}{8} \]
Second part:
\[ \int_{5/4}^{2}(4x-5)\,dx = \left[2x^2 - 5x\right]_{5/4}^{2} = (8 - 10) - \left(\frac{25}{8} - \frac{25}{4}\right) = -2 + \frac{25}{8} = \frac{9}{8} \]
Add:
\[ \frac{25}{8} + \frac{9}{8} = \frac{34}{8} = \frac{17}{4} \]

Final Answer:
The integral equals \(\dfrac{17}{4}\), option (A). \[ \boxed{\frac{17}{4} \text{ (A)}} \]