Question

\(\int _{0}^{2}|4x-5|dx=\)

• A. \(\frac{17}{4}\)
• B. \(\frac{-17}{4}\)
• C. \(\frac{4}{17}\)
• D. \(\frac{-4}{17}\)

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
\(4x-5=0\) at \(x=\dfrac{5}{4}\), which lies in \([0,2]\). So \(|4x-5|=5-4x\) on \([0,\tfrac{5}{4}]\) and \(4x-5\) on \([\tfrac{5}{4},2]\).

Step 2: First Part
\(\int_0^{5/4}(5-4x)dx=\left[5x-2x^{2}\right]_0^{5/4}=\dfrac{25}{4}-\dfrac{25}{8}=\dfrac{25}{8}\).

Step 3: Second Part
\(\int_{5/4}^{2}(4x-5)dx=\left[2x^{2}-5x\right]_{5/4}^{2}=-2-\left(-\dfrac{25}{8}\right)=\dfrac{9}{8}\).

Step 4: Add
\(\dfrac{25}{8}+\dfrac{9}{8}=\dfrac{34}{8}=\dfrac{17}{4}\), option (A).

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