Mathematics Compilations

Let us look at one of the most easiest problems in calculus $$\int_0^{\frac{\pi}{2}}\sqrt{4\sin^2\frac{x}{2}-4\sin\frac{x}{2}+1}\; dx$$ Seems pretty simple and straightforward, doesn't it? Inside the square root, the expression evaluates to $(2\sin\frac{x}{2}-1)^2$, and thus the integral will be $$\int_0^{\frac{\pi}{2}}\Bigl(2\sin\frac{x}{2}-1\Bigr)dx$$ This then can be evaluated by the following: $$\biggl[-4\cos\frac{x}{2} -x\biggr]_0^{\frac{\pi}{2}}$$ Hold on for the surprise... Its wrong! What seemed like a really straightforward and easy problem, is designed in such a way to appear easy so that we approach the wrong way. Why is the approach wrong? We did not consider the fact that a simple square root symbol represents its principal square root i.e., the positive square root. That means each and every value the expression inside the original integral evaluates to a positive quantity. But that is not the case when we simplified the integral to  $$\int...