Question

Tính tích phân: \(\int_{\dfrac{1}{8}}^{\dfrac{1}{3}}\dfrac{1}{x}\sqrt{\dfrac{1+x}{x}}dx\)

Answer 1

Đặt \(\sqrt{\dfrac{1+x}{x}}=t\Leftrightarrow\dfrac{1}{x}=t^2-1\Rightarrow x=\dfrac{1}{t^2-1}\Rightarrow dx=-\dfrac{2t}{\left(t^2-1\right)}dt\)

\(I=\int\limits^2_3\left(t^2-1\right).t.\left(\dfrac{-2t}{\left(t^2-1\right)^2}\right)dt=\int\limits^3_2\dfrac{2t^2}{t^2-1}dt=\int\limits^3_2\left(2+\dfrac{2}{t^2-1}\right)dt\)

\(=\left(2t+ln\left|\dfrac{t-1}{t+1}\right|\right)|^3_2=...\)