Tích phân \(I = \int\limits...

B

- Hướng dẫn giải

\(\begin{array}{l}I = \int\limits_0^a {x\sqrt {x + 1} } dx\\ = \int\limits_0^a {\left( {x + 1} \right)\sqrt {x + 1} } dx - \int\limits_0^a {\sqrt {x + 1} } dx\\ = \int\limits_0^a {{{\left( {x + 1} \right)}^{\frac{3}{2}}}} dx - \int\limits_0^a {{{\left( {x + 1} \right)}^{\frac{1}{2}}}} dx\\{\rm{ = }}\left. {\left[ {\frac{2}{5}{{\left( {x + 1} \right)}^{\frac{5}{2}}}} \right]} \right|_0^a - \left. {\left[ {\frac{2}{3}{{\left( {x + 1} \right)}^{\frac{3}{2}}}} \right]} \right|_0^a\\{\rm{ = }}\frac{2}{5}\sqrt {{{\left( {x + 1} \right)}^5}} - \frac{2}{3}\sqrt {{{\left( {x + 1} \right)}^3}} + \frac{4}{{15}}\end{array}\)