Question

\(\int_0^1x\sqrt{1-x}dx\)

Answer 1

Đặt \(\sqrt{1-x}=u\Rightarrow x=1-u^2\Rightarrow dx=-2udu\)

\(\left\{{}\begin{matrix}x=0\Rightarrow u=1\\x=1\Rightarrow u=0\end{matrix}\right.\)

\(\Rightarrow I=\int\limits^0_1\left(1-u^2\right).u.\left(-2udu\right)=\int\limits^1_0\left(2u^2-2u^4\right)du=\left(\dfrac{2}{3}u^3-\dfrac{2}{5}u^5\right)|^1_0=\dfrac{4}{15}\)