Question

$\int\limits_{0}^{1} \left|5x -3\right|dx $ is equal to

• A. $\frac{10}{13}$
• B. $\frac{31}{10}$
• C. $\frac{13}{10}$
• D. None of these

Answer 1

Answer: (C)

$I=\int_{0}^{1}|5 x-3| d x$
Now, $|5 x-3|=\begin{cases} 5 x-3 & ; x \geq \frac{3}{5} \\-(5 x-3) & ; x<\frac{3}{5} \end{cases}$
$\therefore I=\int_{0}^{1}|5 x-3| d x=\int_{0}^{3 / 5}-(5 x-3) d x +\int_{3 / 5}^{1}(5 x-3) d x$
$=-\left[\frac{5 x^{2}}{2}-3 x\right]_{0}^{1 / 5}+\left[\frac{5 x^{2}}{2}-3 x^{1}\right.$
$=-\left[\left(\frac{5}{2} \times \frac{9}{25}-\frac{9}{5}\right)-(0-0)\right]+\left[\left(\frac{5}{2}-3\right)\right.$
$\left.-\left(\frac{5}{2} \times \frac{9}{25}-\frac{9}{5}\right)\right]$
$=-\frac{45}{50}+\frac{9}{5}-\frac{1}{2}-\frac{45}{50}+\frac{9}{5}=\frac{13}{10}$