Tính: \(\displaystyle \int {(2x - 3)\sqrt {x - 3} dx}\)
Hãy suy nghĩ và trả lời câu hỏi trước khi xem đáp án
Lời giải:
Báo saiĐặt \(\displaystyle u = \sqrt {x - 3} \)\(\displaystyle \Rightarrow {u^2} = x - 3 \Rightarrow 2udu = dx\)
\(\displaystyle \Rightarrow \int {(2x - 3)\sqrt {x - 3} dx} \) \(\displaystyle = \int {\left[ {2\left( {{u^2} + 3} \right) - 3} \right].u.2udu} \) \(\displaystyle = 2\int {{u^2}\left( {2{u^2} + 3} \right)du} \) \(\displaystyle = 2\int {\left( {2{u^4} + 3{u^2}} \right)du} \)
\(\displaystyle = 2\left( {2.\frac{{{u^5}}}{5} + 3.\frac{{{u^3}}}{3}} \right) + C\)
\(\displaystyle = \frac{4}{5}{u^5} + 2{u^3} + C\)
\(\displaystyle = \frac{4}{5}.{\left( {\sqrt {x - 3} } \right)^5} + 2{\left( {\sqrt {x - 3} } \right)^3} + C\)
\(\displaystyle = \frac{4}{5}{\left( {x - 3} \right)^{\frac{5}{2}}} +2 {\left( {x - 3} \right)^{\frac{3}{2}}} + C\)
