[beginner01]

$\displaystyle\int_{1}^{3}\left(\frac{2}{x}+\sqrt{\frac{4}{x}-1}\right)^{-\frac{1}{2}}dx$
=$\displaystyle\int_{1}^{3}\sqrt{2}\left(\frac{4}{x}-1+2\sqrt{\frac{4}{x}-1}+1\right)^{-\frac{1}{2}}dx$
=$\displaystyle\int_{1}^{3}\sqrt{2}\frac{1}{\sqrt{\frac{4}{x}-1}+1}dx$
=$\displaystyle\sqrt{2}\int_{1}^{3}\frac{1}{\sqrt{\frac{4}{x}-1}+1}dx$
ตรงนี้ไม่เข้าใจว่าได้มายังไงเหมือนกันนะครับ แต่ไปเช็คจาก mathematica ก็ได้ว่า
=$\displaystyle\sqrt{2}\left[\frac{1}{2}\left(-x\sqrt{\frac{4}{x}-1}+x+2\left(\ln\left(x\sqrt{\frac{4}{x}-1}+2\right)\right)\right)\right]^3_1$
=$\sqrt{2}$