Animated Solution for Mathematics - Definite Integration: The integral ${\int }_{\pi /6}^{\pi /4}\frac{dx}{\sin 2x({\tan }^{5}x+{\cot }^{5}x)}$ equals:

The Art of Simplifying Trigonometric Integrals. Welcome, aspirant. Today, we are going to tackle a problem that, at first glance, might seem like a chaotic mess of trigonometric functions.

You see an integral like:

In the world of JEE Advanced, intimidation is just a test of your patience and your toolkit. Let's break this down together.

The Transformation

The key to solving complex trigonometric integrals is to reduce the number of different functions involved. We have $\sin 2x$, $\tan x$, and $\cot x$. Our goal is to bring everything into a single family, preferably $\tan x$ and $\sec x$.

First, we use the double angle identity: $\sin 2x=2\sin x\cos x$. Next, we know that ${\cot }^{5}x=\frac{1}{{\tan }^{5}x}$. Substituting these into our integral, we get:

Now, let's simplify the denominator. The term $({\tan }^{5}x+\frac{1}{{\tan }^{5}x})$ becomes $\frac{{\tan }^{10}x+1}{{\tan }^{5}x}$. Moving the ${\tan }^{5}x$ to the numerator, we have:

The Hidden Pattern

We still have $\sin x\cos x$ in the denominator. If we recognize that $\frac{1}{\sin x\cos x}=\frac{{\sec }^{2}x}{\tan x}$, the integral transforms beautifully.

Substituting this, we get:

Simplifying the powers of $\tan x$, we get $\frac{{\tan }^{5}x}{\tan x}={\tan }^{4}x$. Since ${\tan }^{10}x=({\tan }^{5}x{)}^2$, our integral becomes:

Do you see it now? The derivative of ${\tan }^{5}x$ is $5{\tan }^{4}x{\sec }^{2}xdx$. It is sitting right there in the numerator!

The Substitution

Let $t={\tan }^{5}x$. Then $dt=5{\tan }^{4}x{\sec }^{2}xdx$, which means ${\tan }^{4}x{\sec }^{2}xdx=\frac{dt}{5}$.

Now, we must change the limits: When $x=\frac{\pi }{6}$, $t=(\tan \frac{\pi }{6}{)}^5=(\frac{1}{\sqrt{3}}{)}^5=\frac{1}{9\sqrt{3}}$. When $x=\frac{\pi }{4}$, $t=(\tan \frac{\pi }{4}{)}^5=1^5=1$.

Our integral is now:

This is a standard integral. The integral of $\frac{1}{t^2+1}$ is ${\tan }^{-1}t$.

Applying the limits:

Since ${\tan }^{-1}(1)=\frac{\pi }{4}$, our final answer is:

The beauty of this problem lies in how a seemingly impossible expression collapses into a simple standard form through careful, logical steps. Keep practicing, and you will start to see these patterns everywhere.