Animated Solution for Mathematics - Definite Integration: The value of ${\int }_{-\pi /2}^{\pi /2}\frac{{\sin }^{2}x}{1+2^x}dx$ is :

Welcome, fellow traveler on the JEE journey. Today, we are going to dismantle a problem that, at first glance, looks like a nightmare.

We are looking at the integral:

That $2^x$ in the denominator is designed to intimidate you. It is designed to make you think you need complex substitution or some obscure integration technique.

But here is the secret: in JEE Advanced, when you see a complicated function trapped between symmetric limits like $[-\frac{\pi }{2},\frac{\pi }{2}]$, you are not looking at a calculation problem; you are looking at a symmetry problem.

Your Secret Weapon

Whenever you see limits of the form $[a,b]$, the King's Property is your best friend. It states that:

In our case, $a=-\frac{\pi }{2}$ and $b=\frac{\pi }{2}$. Their sum is $a+b=0$. This means our property simplifies to replacing $x$ with $-x$.

Let's see what happens when we do this. We define our integral as:

Applying the property, we get:

Since ${\sin }^2(-x)={\sin }^{2}x$, our integral becomes:

Now, let's look at that denominator: $1+2^{-x}$. We know that $2^{-x}=\frac{1}{2^x}$.

So, the denominator is:

When we substitute this back into our integral, the $2^x$ in the denominator of the denominator flips up to the numerator. We get:

This is our second equation. Now, watch the magic. If we add our original integral $I$ and this new version, we get:

Look at the numerators! We have ${\sin }^{2}x+2^x{\sin }^{2}x$. If we factor out the ${\sin }^{2}x$, we get ${\sin }^{2}x(1+2^x)$.

And look at the denominator: $1+2^x$. They cancel out perfectly! We are left with:

The exponential term, which seemed so daunting, has vanished into thin air. This is the elegance of mathematics.

Since ${\sin }^{2}x$ is an even function, we can write:

We cannot integrate ${\sin }^{2}x$ directly, so we use the half-angle identity: ${\sin }^{2}x=\frac{1-\cos 2x}{2}$.

Our integral becomes:

Evaluating this at the limits, we get:

And there it is! The final answer is $\frac{\pi }{4}$. Remember, the next time you face a 'monster' integral, don't panic. Look for the symmetry, apply the property, and trust the algebra.