Request for proof of the following identity.

Question

How do we establish the following identity? \begin{align} &\int_{0}^{\frac{\pi}{2}} \frac{1}{(\sin(\theta) + \cos(\theta)) \sqrt{\sin(\theta) \cos(\theta)}} \,d{\theta} \\ \stackrel{\text{def}}{=} &\lim_{a \rightarrow 0^{+} \,,\, b \rightarrow (\frac{\pi}{2})^{-}} \int_{a}^{b} \frac{1}{(\sin(\theta) + \cos(\theta)) \sqrt{\sin(\theta) \cos(\theta)}} \,d{\theta} \\ = &\pi. \end{align}

Answer 1

If we recall that $\sin(\theta)=\tan(\theta)\cos(\theta)$ then:

$$I=\int_{0}^{\frac{\pi}{2}} \frac{1}{(1 + \tan(\theta)) \sqrt{\tan(\theta)}} \frac{\,d{\theta}}{\cos^2(\theta)}$$

$$=\int_{0}^{\frac{\pi}{2}} \frac{1}{(1 + \tan(\theta)) \sqrt{\tan(\theta)}} \,d[\tan(\theta)]$$

$$=\int_{0}^{\infty} \frac{1}{(1 + z) \sqrt{z}} \,dz=\pi$$