Animated Solution for Mathematics - Indefinite Integration: $\int \frac{dx}{\cos x+\sqrt{3}\sin x}$ equals

Mastering the Linear Combination

Welcome, fellow traveler on the path to JEE mastery. Today, we are going to demystify one of the most elegant techniques in calculus: the transformation of a linear combination of sine and cosine.

When you first look at the integral $I=\int \frac{dx}{\cos x+\sqrt{3}\sin x}$, it might seem like a standard, perhaps even boring, problem. But beneath the surface lies a beautiful geometric reality that, once understood, will change how you approach trigonometric integrals forever.

The Geometric Intuition

Whenever you encounter an expression of the form $a\cos x+b\sin x$, do not panic. Instead, think of it as a vector.

Imagine a right-angled triangle where the base is $b=\sqrt{3}$ and the height is $a=1$. The hypotenuse of this triangle, which we call $R$, is given by the Pythagorean theorem:

This $R$ is our scaling factor. By multiplying and dividing our denominator by $2$, we force the expression into a form that mirrors the compound angle identity $\sin (A+B)=\sin A\cos B+\cos A\sin B$.

The Transformation

We rewrite the denominator as:

Now, look closely at the terms inside the bracket. We know that $\sin (\frac{\pi }{6})=\frac{1}{2}$ and $\cos (\frac{\pi }{6})=\frac{\sqrt{3}}{2}$.

Substituting these values, the expression becomes:

This is the exact expansion of $\sin (x+\frac{\pi }{6})$. Our denominator has been tamed; it is now simply $2\sin (x+\frac{\pi }{6})$.

The Final Integration

With the denominator simplified, our integral becomes:

I know that seeing a cosecant integral might make you nervous, but remember the standard formula: $\int \csc udu=\log |\tan (\frac{u}{2})|+C$. Here, our $u$ is $x+\frac{\pi }{6}$, and therefore $\frac{u}{2}$ becomes $\frac{x}{2}+\frac{\pi }{12}$.

Applying the formula, we arrive at our final, elegant result:

The Beauty of the Method

This problem is a perfect example of why we study mathematics. It is not just about memorizing formulas; it is about recognizing patterns and using geometry to simplify the complex.

By transforming a sum of two functions into a single function, we turned a potentially messy integration into a simple, standard form. Keep this geometric triangle analogy in your toolkit—it will serve you well in many more JEE problems to come.