Evaluate int_3^7 xsqrt(x-3) dx 

Explanation:

Another Explanation (5): সমাধান: 🤔 ধরি, \(x-3 = u\) সুতরাং \(x = u+3\) এবং \(dx = du\) যখন \(x = 3\), তখন \(u = 3-3 = 0\) যখন \(x = 7\), তখন \(u = 7-3 = 4\) অতএব, \(\int_3^7 x\sqrt{x-3} \, dx = \int_0^4 (u+3)\sqrt{u} \, du\) \( = \int_0^4 (u^{3/2} + 3u^{1/2}) \, du\) \( = \left[ \frac{u^{5/2}}{5/2} + 3\frac{u^{3/2}}{3/2} \right]_0^4\) \( = \left[ \frac{2}{5} u^{5/2} + 2 u^{3/2} \right]_0^4\) \( = \left( \frac{2}{5} (4)^{5/2} + 2 (4)^{3/2} \right) - \left( \frac{2}{5} (0)^{5/2} + 2 (0)^{3/2} \right)\) \( = \frac{2}{5} (2^2)^{5/2} + 2 (2^2)^{3/2}\) \( = \frac{2}{5} (2^5) + 2 (2^3)\) \( = \frac{2}{5} \cdot 32 + 2 \cdot 8\) \( = \frac{64}{5} + 16\) \( = \frac{64 + 80}{5}\) \( = \frac{144}{5}\) সুতরাং, \(\int_3^7 x\sqrt{x-3} \, dx = \frac{144}{5}\) 🎉