Animated Solution for Mathematics - Indefinite Integration: The integral $\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{1+\frac{2}{\sqrt{3}}\sin 2x}dx$ is equal to

Welcome, fellow traveler on the path to JEE excellence. Today, we are not just solving an integral; we are performing a surgical operation on a complex trigonometric expression.

When you first look at the integral

it is natural to feel a sense of intimidation. It looks cluttered, messy, and frankly, quite uninviting. But remember, in the world of advanced calculus, complexity is often just a mask for a hidden, elegant symmetry.

Clearing the Fog

The first step is to simplify our environment. Those fractions involving $\sqrt{3}$ are not there to annoy you; they are signposts.

By multiplying the numerator and denominator by $\sqrt{3}$, we transform the expression into

Now, let us divide both the numerator and the denominator by $2$. This is a classic move—whenever you see coefficients like $\sqrt{3}$ and $1$, think of the $30^{\circ }-60^{\circ }-90^{\circ }$ triangle.

We get

Notice that $\frac{\sqrt{3}}{2}$ is exactly $\sin (\frac{\pi }{3})$. Our denominator is now $\sin (\frac{\pi }{3})+\sin (2x)$.

The Beauty of Sum-to-Product

Now, we invoke the power of the sum-to-product identity:

Applying this to our denominator, we get

This is the turning point! We have successfully factored the denominator into two distinct trigonometric functions.

But what about the numerator? We need to express $\left(\frac{\sqrt{3}-1}{2}\right)(\cos x-\sin x)$ in a way that matches these factors. By expanding the terms, we discover the magic: the numerator is equivalent to

The Grand Cancellation

Now, watch closely as the complexity collapses. Substituting our new numerator and denominator back into the integral, we have

By splitting this into two separate integrals, the terms cancel out beautifully:

We are left with the integrals of $\csc$ and $\sec$.

The Final Integration

Using the standard results, the integral of $\csc \theta$ is $\ln |\tan (\frac{\theta }{2})|$ and the integral of $\sec \theta$ is $\ln |\tan (\frac{\theta }{2}+\frac{\pi }{4})|$. Applying these, we obtain:

Simplifying the second argument gives us $\frac{x}{2}+\frac{\pi }{6}$. Finally, using the property of logarithms $\ln A-\ln B=\ln (\frac{A}{B})$, we arrive at our destination:

This is the elegance of mathematics. We started with a chaotic expression and, through logical steps and the application of fundamental identities, arrived at a clean, precise result. Never fear the complexity; trust the process, and the math will always reveal its hidden order.