Animated Solution for Mathematics - Definite Integration: ${\int }_6^{16}\frac{{\log }_e{x}^2}{{\log }_e{x}^2+{\log }_e(x^2-44x+484)}dx$ is equal to:

Welcome, fellow traveler on the path to JEE excellence. Today, we are not just solving an integral; we are uncovering a hidden symmetry.

When you first look at the expression

it might seem intimidating. It looks like a complex mess of logarithms and quadratics, but in the world of competitive mathematics, complexity is often just a mask for elegance.

Let us look at the denominator. Specifically, focus your attention on the quadratic expression $x^2-44x+484$.

If you have been practicing your algebraic identities, you might notice that $484$ is the square of $22$, and $44$ is exactly $2\times 22$. This is the classic expansion of $(x-22{)}^2$.

By recognizing this, we transform our integral into:

The King's Property states that for any continuous function $f(x)$ on the interval $[a,b]$, the integral ${\int }_a^{b}f(x)dx$ is equal to ${\int }_a^{b}f(a+b-x)dx$.

Here, our limits are $6$ and $16$. Their sum is $22$. This is not a coincidence; it is a mathematical invitation.

By replacing $x$ with $(22-x)$, we are essentially looking at the function from the other side of the mirror. Let us perform the substitution:

Look closely at the second term in the denominator. The expression $((22-x)-22{)}^2$ simplifies beautifully to $(-x{)}^2$, which is just $x^2$.

Our new integral is:

Now, we have two versions of $I$. If we add them together, something magical happens.

The denominators are identical, and the numerators add up to the exact same expression as the denominator:

Because the numerator and denominator are now the same, the entire integrand collapses into $1$. We are left with the simplest integral imaginable:

Evaluating this is straightforward:

Therefore, the final result is:

We started with a complex logarithmic expression, and through the power of symmetry and the King's Property, we reduced it to a simple subtraction. This is the beauty of JEE mathematics.

It is not about brute force; it is about finding the right perspective. Whenever you face a difficult integral, pause, look for the symmetry, and look for the perfect squares. The answer is often hiding in plain sight, waiting for you to apply the right property.