Animated Solution for Mathematics - Definite Integration: The value of ${\int }_{\sqrt{\ln 2}}^{\sqrt{\ln 3}}\frac{x\sin x^2}{\sin x^2+\sin (\ln 6-x^2)}dx$ is

The given integral is:

It is easy to feel overwhelmed, but in the world of JEE Advanced, complexity is often just a mask for elegance. Let us peel back that mask together.

The presence of $x$ outside and $x^2$ inside the sine functions is a classic signal. Whenever you see a function of $x^2$ paired with an $x$, your intuition should immediately scream substitution.

Let us set $t=x^2$. Differentiating both sides, we get $2xdx=dt$, or $xdx=\frac{1}{2}dt$.

We must also update our boundaries. When $x=\sqrt{\ln 2}$, $t=\ln 2$. When $x=\sqrt{\ln 3}$, $t=\ln 3$.

Our integral now breathes a sigh of relief, transforming into:

This is our first equation, let us call it $(1)$.

Now, we encounter one of the most powerful tools in our arsenal: the King's Property. It states that:

This property allows us to swap the variable $t$ with the sum of the limits minus $t$ without changing the area under the curve. Here, $a=\ln 2$ and $b=\ln 3$, so $a+b=\ln 6$.

Let us replace $t$ with $(\ln 6-t)$ in our integral. The denominator becomes $\sin (\ln 6-t)+\sin (\ln 6-(\ln 6-t))$, which simplifies beautifully to $\sin (\ln 6-t)+\sin t$.

The denominator remains unchanged. This gives us a second form of our integral:

Let us call this equation $(2)$.

Now, watch the magic happen. If we add equation $(1)$ and equation $(2)$, we get $2I$ on the left side.

On the right side, because the denominators are identical, we simply add the numerators:

The entire complex fraction cancels out to become $1$. We are left with the incredibly simple task of integrating $1$ with respect to $t$:

The integral of $1$ is just $t$. Evaluating this from $\ln 2$ to $\ln 3$, we get $(\ln 3-\ln 2)$, which is $\ln \frac{3}{2}$.

Finally, dividing by $2$, we arrive at our destination:

It is a beautiful result. What started as a daunting, complex expression resolved into a simple logarithmic value. This is the essence of mathematics: finding the hidden symmetry that turns a nightmare into a dream.