Q.∫ (from π/6 to π/3) √(cos x) / (√(sin x) + √(cos x)) dx is equal to:

The King Property of Definite Integrals

Walk a path from $a$ to $b$ measuring something at each step; now walk it backwards from $b$ to $a$. The King Property says the total is unchanged — provided you also reverse how you measure. It is one of the most useful shortcuts for definite integrals.

The limits stay $a$ to $b$; only the argument changes, $x \to a+b-x$.

Where it comes from

Substitute $t = a + b - x$, so $dx = -dt$; when $x=a$, $t=b$ and when $x=b$, $t=a$:

Renaming $t$ back to $x$ gives the result. So it is not a trick — just substitution.

Why it helps

Adding the original integral to its "mirror" often collapses the integrand. For instance, with $I = \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx$, the property replaces $\sin x$ by $\cos x$ (since $\sin(\tfrac{\pi}{2}-x)=\cos x$). Adding the two forms: