Animated Solution for Mathematics - Definite Integration: Evaluate ${\int }_0^{\pi }\frac{x\sin 2x\sin \left(\frac{\pi }{2}\cos x\right)}{2x-\pi }dx$

Imagine you are standing before a complex, intimidating integral. It looks like a monster, doesn't it?

But in the world of JEE Advanced, every monster has a weakness. Our weakness here is the denominator, $2x-\pi$. Whenever you see a linear term in the denominator combined with bounds like $[0,\pi ]$, your intuition should immediately scream: "King's Property!"

The King's Property

The King's Property, ${\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x)dx$, is not just a formula; it is a lens that reveals the hidden symmetry of a function. Let us define our integral as $I$.

By replacing $x$ with $\pi -x$, we are essentially flipping the function across the midpoint of the interval. Let's watch the transformation:

The numerator $\sin 2x$ becomes $\sin 2(\pi -x)=\sin (2\pi -2x)=-\sin 2x$.

The term $\sin (\frac{\pi }{2}\cos x)$ transforms into $\sin (\frac{\pi }{2}\cos (\pi -x))=\sin (-\frac{\pi }{2}\cos x)=-\sin (\frac{\pi }{2}\cos x)$.

The Beautiful Cancellation

Now, look at the denominator. Replacing $x$ with $\pi -x$ gives $2(\pi -x)-\pi =\pi -2x$.

This is the magic moment. Our new integral $I$ now has a denominator of $\pi -2x$, which is just $-(2x-\pi )$.

When we add the original $I$ to our transformed $I$, we get:

By factoring out the negative sign from the second denominator, we get a common denominator of $2x-\pi$. The numerators combine to $x+(x-\pi )=2x-\pi$.

They cancel perfectly! We are left with:

The monster has been tamed.

The Transformation

We are now left with $I=\frac{1}{2}{\int }_0^{\pi }\sin 2x\sin (\frac{\pi }{2}\cos x)dx$. Using the identity $\sin 2x=2\sin x\cos x$, the $2$ cancels with the $\frac{1}{2}$, leaving:

This structure is a perfect candidate for $u$-substitution. Let $u=\frac{\pi }{2}\cos x$.

Then $du=-\frac{\pi }{2}\sin xdx$, which means $\sin xdx=-\frac{2}{\pi }du$. As we change the limits, $x=0$ maps to $u=\frac{\pi }{2}$, and $x=\pi$ maps to $u=-\frac{\pi }{2}$.

The Final Dance

Substituting these into our integral, we get:

We use Integration by Parts on $\int u\sin udu$. Let $f=u$ and $g^{\prime }=\sin u$. The result is $-u\cos u+\sin u$.

Evaluating this from $-\frac{\pi }{2}$ to $\frac{\pi }{2}$ yields $1-(-1)=2$.

Finally, multiplying by our constant $\frac{4}{{\pi }^2}$, we arrive at our elegant answer: $\frac{8}{{\pi }^2}$. It is a journey from chaos to order, and that is the essence of physics and mathematics.