ʃ₀¹⁰ [x−5] dx = ?

প্রশ্ন: \( \int_{0}^{10} |x-5| \, dx = ? \)

সমাধান:

আমরা জানি, \( |x-5| = \begin{cases} 5-x, & \text{যদি } x < 5 \\ x-5, & \text{যদি } x \geq 5 \end{cases} \)

অতএব,

\( \int_{0}^{10} |x-5| \, dx = \int_{0}^{5} (5-x) \, dx + \int_{5}^{10} (x-5) \, dx \)

প্রথম ইন্টিগ্রাল:

\( \int_{0}^{5} (5-x) \, dx = \left[ 5x - \frac{x^2}{2} \right]_{0}^{5} = \left( 5(5) - \frac{5^2}{2} \right) - \left( 5(0) - \frac{0^2}{2} \right) = 25 - \frac{25}{2} = \frac{50-25}{2} = \frac{25}{2} \)

দ্বিতীয় ইন্টিগ্রাল:

\( \int_{5}^{10} (x-5) \, dx = \left[ \frac{x^2}{2} - 5x \right]_{5}^{10} = \left( \frac{10^2}{2} - 5(10) \right) - \left( \frac{5^2}{2} - 5(5) \right) = (50 - 50) - \left( \frac{25}{2} - 25 \right) = 0 - \left( \frac{25 - 50}{2} \right) = - \left( -\frac{25}{2} \right) = \frac{25}{2} \)

সুতরাং,

\( \int_{0}^{10} |x-5| \, dx = \frac{25}{2} + \frac{25}{2} = \frac{50}{2} = 25 \)

অতএব, উত্তর: 25 🎉