[FranceZii Siriseth]

$$ \int \sqrt{\frac{x+1}{x+3}}\ dx $$
$$ 2\int \sqrt{x+1}\ d(\sqrt{x+3})$$
$$ 2(\sqrt{(x+1)(x+3)}-\int \sqrt{x+3}\ d(\sqrt{x+1})) $$

พิจารณา $$\int \sqrt{x+3}\ d(\sqrt{x+1}) $$
$$=\frac{1}{2}\int \sqrt{\frac{x+3}{x+1}}\ dx $$
$$=\frac{1}{2}\int \sqrt{1+\frac{2}{x+1}}\ dx $$

ให้ $\frac{2}{x+1}= tan(\theta )^2$
$$\frac{-2}{(x+1)^2}\ dx= 2tan(\theta )sec(\theta)^2\ d\theta$$

$$=\frac{1}{2}\int \sqrt{1+\frac{2}{x+1}}\ dx $$
$$=\frac{1}{2}\int sec(\theta)2tan(\theta )sec(\theta)^2\ \frac{(x+1)^2}{-2} d\theta$$
$$=\frac{1}{2}\int sec(\theta)2tan(\theta )sec(\theta)^2\ \frac{-2}{tan(\theta)^4}d\theta$$
$$=-2\int \frac{sec(\theta)^3}{tan(\theta)^3}\ d\theta$$
$$=-2\int csc(\theta)^3 d\theta$$

ที่เหลือก็ By Parts ปะครับ