f(x) = |x| হলে  int_-1^1 f(x)dx =? 

প্রশ্ন: \(f(x) = |x|\) হলে \(\int_{-1}^{1} f(x) dx = ?\)

আমরা জানি, \(|x| = \begin{cases} -x, & \text{যদি } x < 0 \\ x, & \text{যদি } x \geq 0 \end{cases}\)

সুতরাং, \(\int_{-1}^{1} |x| dx\) কে দুইটি অংশে ভাগ করে লেখা যায়:

\(\int_{-1}^{1} |x| dx = \int_{-1}^{0} |x| dx + \int_{0}^{1} |x| dx\)

\(= \int_{-1}^{0} (-x) dx + \int_{0}^{1} x dx\)

\(= \left[-\frac{x^2}{2}\right]_{-1}^{0} + \left[\frac{x^2}{2}\right]_{0}^{1}\)

\(= \left(-\frac{0^2}{2} - \left(-\frac{(-1)^2}{2}\right)\right) + \left(\frac{1^2}{2} - \frac{0^2}{2}\right)\)

\(= \left(0 + \frac{1}{2}\right) + \left(\frac{1}{2} - 0\right)\)

\(= \frac{1}{2} + \frac{1}{2}\)

\(= 1\)

অতএব, \(\int_{-1}^{1} |x| dx = 1\) 🥳