Animated Solution for Mathematics - Indefinite Integration: The integral $\int {\left(\frac{x}{x\sin x+\cos x}\right)}^{2}dx$ is equal to (where $C$ is a constant of integration) :

Welcome, future engineer. Today, we stand before a problem that, at first glance, might make your heart skip a beat. We are looking at the integral:

It is a complex, trigonometric, algebraic beast. But remember, in the arena of JEE Advanced, intimidation is often just a mask for elegance. Let us peel back the layers of this problem together.

Analyzing the Denominator

Whenever you see a complex algebraic and trigonometric mix in the denominator, your first instinct should be to investigate it. Do not rush to integrate; instead, pause.

Look at the denominator $f(x)=x\sin x+\cos x$. Let us differentiate it with respect to $x$:

The $\sin x$ terms cancel out perfectly, leaving us with $x\cos x$. This is our golden key—the hidden structure that will unlock the entire problem.

Creating the Derivative

Now that we know the derivative of the denominator is $x\cos x$, we look back at our numerator, which is $x^2$. We need to create $x\cos x$ in the numerator to make the integration possible.

We split $x^2$ into $x\cdot x$ and strategically multiply and divide the entire expression by $\cos x$:

This brilliant manipulation transforms our integral into a product of two distinct functions, setting the stage perfectly for Integration by Parts.

We apply the formula $\int udv=uv-\int vdu$. We choose $u=x\sec x$ because it is relatively easy to differentiate.

That leaves the rest of the expression as our second part:

We chose this as $dv$ specifically because we engineered its numerator to be the exact derivative of the term inside the denominator's square.

First, we find $du$ by differentiating $u=x\sec x$:

Next, we find $v$ by integrating $dv$. Substituting $t=x\sin x+\cos x$, we get $dt=x\cos xdx$:

Now, we piece everything together using the formula $uv-\int vdu$:

Inside the integral, we convert to basic sines and cosines:

Notice that the numerator contains the exact same expression as our main denominator: $x\sin x+\cos x$. These two terms cancel each other out completely:

Finally, we combine all our pieces to write the final answer:

The key takeaway here is to always look for hidden derivatives. By manipulating the numerator to match the derivative of the denominator, we unlocked the entire problem. Keep practicing this technique, and you will find that even the most intimidating integrals are just puzzles waiting to be solved.