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

Welcome, fellow traveler on the path to JEE mastery. Today, we are going to demystify an integral that often leaves students feeling stuck:

At first glance, it looks deceptively simple, yet it lacks the obvious substitution that makes life easy. This is where the true beauty of calculus shines—not in brute force, but in strategic manipulation.

Seeing the Unseen

Whenever you encounter a rational trigonometric function where the numerator and denominator are linear combinations of $\sin x$ and $\cos x$, there is a powerful, almost magical, strategy. We aim to express the numerator as a linear combination of the denominator and its derivative.

Let our denominator be $D=\sin x-\cos x$. If we differentiate this, we get:

This $D^{\prime }$ is the key to unlocking the integral.

Our numerator is just $\sin x$. To make it look like $A(D)+B(D^{\prime })$, we need a bit more flexibility. We multiply and divide the integral by $2$, giving us:

Now, we rewrite $2\sin x$ as $\sin x+\sin x$. To introduce our $D$ and $D^{\prime }$, we add and subtract $\cos x$ in the numerator:

Rearranging this, we get $(\sin x-\cos x)+(\sin x+\cos x)$, which is exactly $D+D^{\prime }$.

Now, substitute this back into our integral:

Splitting this into two fractions, we get:

The first term simplifies beautifully to $1$. The second term is now in the form $\int \frac{f^{\prime }(x)}{f(x)}dx$, which integrates to $\ln |f(x)|$.

Thus, the integral becomes:

Integrating both parts, we arrive at our final, polished result:

Take a moment to appreciate the elegance of this solution. We didn't just solve an integral; we transformed a complex expression into a sum of simple, fundamental functions. This is the essence of JEE mathematics—finding the hidden structure and letting it guide you to the answer.