\( \int_{0}^{1}\displaystyle|5x-3|\,dx= \)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{13}{10} \)
The modulus changes sign at \( x=\frac{3}{5} \), so the integral splits there:
\( \displaystyle \int_{0}^{3/5}(3-5x)\,dx=\frac{9}{10} \)
\( \displaystyle \int_{3/5}^{1}(5x-3)\,dx=\frac{2}{5} \)
Adding,
\( \displaystyle \frac{9}{10}+\frac{2}{5}=\frac{13}{10} \)
Shortcut: The graph of \( |5x-3| \) on [0, 1] makes two triangles:
\( \displaystyle \frac{1}{2}\cdot\frac{3}{5}\cdot 3+\frac{1}{2}\cdot\frac{2}{5}\cdot 2=\frac{13}{10} \)