Animated Solution for Mathematics - Indefinite Integration: $\int \frac{dx}{\cos x-\sin x}$ is equal to

Welcome, future engineer! Today, we are going to tackle an integral that looks deceptively simple but hides a beautiful geometric secret. We are looking at the integral:

At first glance, you might feel the urge to jump into complex substitutions, but pause. In the world of JEE Advanced, the most powerful tool is often the ability to see the 'hidden' structure of an expression.

Whenever you see a denominator that is a linear combination of sine and cosine, like $a\cos x+b\sin x$, your brain should immediately trigger the 'Harmonic Addition' reflex. We want to collapse these two terms into one.

To do this, we multiply and divide by the amplitude $R=\sqrt{a^2+b^2}$. In our case, $a=1$ and $b=-1$. Calculating the amplitude, we get:

Now, we factor this $\sqrt{2}$ out:

Here is where the elegance of trigonometry shines. We know that $\frac{1}{\sqrt{2}}$ is the value of both $\cos (\frac{\pi }{4})$ and $\sin (\frac{\pi }{4})$. Substituting these into our expression, we get:

Does that look familiar? It is the classic cosine addition formula: $\cos A\cos B-\sin A\sin B=\cos (A+B)$. With $A=x$ and $B=\frac{\pi }{4}$, our denominator collapses into:

Now, our integral looks much friendlier:

Since $\frac{1}{\cos \theta }=\sec \theta$, we have:

We are now in the territory of standard integrals. The integral of $\sec u$ is $\ln |\tan (\frac{u}{2}+\frac{\pi }{4})|+C$. Applying this with $u=x+\frac{\pi }{4}$, we get:

The final step is just a bit of arithmetic to match the options. Let's simplify the argument of the tangent:

Combining the constants $\frac{\pi }{8}+\frac{2\pi }{8}$ gives us $\frac{3\pi }{8}$. Thus, our final result is:

You see? By transforming the expression, we didn't just solve the problem; we revealed its underlying structure. Keep practicing this condensation technique—it is a superpower in your JEE toolkit!