Giải bài 3.15 trang 166 - SBT Đại số và Giải tích lớp 12
\(\int x\,\ \sqrt{x-1}dx\) bằng :
A. \((x-1)^{\frac{5}{2}} +(x-1)^{\frac{3}{2}}+C\)
B. \(\dfrac{2}{15}\left[3(x-1)^{\frac{5}{2}} -5(x-1)^{\frac{3}{2}}\right]+C\)
C. \(\dfrac{2}{15}\left[3(x-1)^{\frac{5}{2}} +5(x-1)^{\frac{3}{2}}\right]+C\)
D. \(\dfrac{1}{15}\left[3(x-1)^{\frac{5}{2}} +5(x-1)^{\frac{3}{2}}\right]+C\)
\(\begin{align} & \int{x\sqrt{x-1}dx}=\int{\left( x-1 \right)\sqrt{x-1}dx}+\int{\sqrt{x-1}dx} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{2}{5}{{\left( x-1 \right)}^{\frac{5}{2}}}+\frac{2}{3}{{\left( x-1 \right)}^{\frac{3}{2}}}+C \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{2}{15}\left[ 3{{\left( x-1 \right)}^{\frac{5}{2}}}+5{{\left( x-1 \right)}^{\frac{3}{2}}} \right]+C \\ \end{align} \)
Chọn đáp án C