Animated Solution for Mathematics - Definite Integration: The value of ${\int }_0^{\pi /2}\frac{dx}{1+{\tan }^{3}x}$ is

Welcome, future engineer. Today, we are going to dismantle a problem that often makes students freeze in their tracks: the integral $I={\int }_0^{\pi /2}\frac{dx}{1+{\tan }^{3}x}$.

At first glance, it looks like a nightmare. You might be tempted to reach for complex substitutions or partial fractions, but I want you to take a deep breath.

In the world of JEE Advanced, when you see an integral with limits from $0$ to $\frac{\pi }{2}$, it is rarely a test of your ability to perform brute-force calculus. It is a test of your ability to see the hidden symmetry. Let us embark on this journey together.

The Setup

First, let us give our integral a name. We call it $I$. Our goal is to find the value of:

The integrand is currently in terms of $\tan x$, which is not the most friendly function to work with. Let us convert everything into the fundamental trigonometric functions, sine and cosine.

We know that $\tan x=\frac{\sin x}{\cos x}$, so ${\tan }^{3}x$ becomes $\frac{{\sin }^{3}x}{{\cos }^{3}x}$. Substituting this into our integral, we get:

Now, let us clean up this complex fraction. By taking the common denominator in the bottom part, we get $\frac{{\cos }^{3}x+{\sin }^{3}x}{{\cos }^{3}x}$.

When we flip the denominator's denominator to the numerator, our integral simplifies beautifully to:

Let us call this Equation 1. This is our starting point.

The King's Property

Now, we invoke the 'King's Property' of definite integrals. This is one of the most powerful tools in your arsenal.

It states that:

Think of this as a horizontal flip of the function. The area under the curve remains unchanged, but the function itself transforms. In our case, $a=\frac{\pi }{2}$.

So, we replace every $x$ in our integrand with $(\frac{\pi }{2}-x)$. The integral becomes:

Using the co-function identities, we know that $\cos (\frac{\pi }{2}-x)=\sin x$ and $\sin (\frac{\pi }{2}-x)=\cos x$. Our integral transforms into:

Let us call this Equation 2.

The Climax

Here is the brilliant part. What happens if we add Equation 1 and Equation 2 together?

On the left side, $I+I=2I$. On the right side, because the denominators are identical, we can simply add the numerators:

The entire fraction cancels out perfectly, leaving us with:

We have turned a terrifying trigonometric integral into the simplest integral possible! The anti-derivative of $1$ is just $x$. Evaluating this from $0$ to $\frac{\pi }{2}$, we get:

Finally, we divide both sides by $2$ to isolate $I$, giving us:

This is the elegance of mathematics. We didn't need complex integration techniques; we needed symmetry.

And remember, this is a generalization: for any real power $n$, ${\int }_0^{\pi /2}\frac{dx}{1+{\tan }^{n}x}=\frac{\pi }{4}$. Keep this in your toolkit, and you will be ready for whatever the JEE throws at you.