Animated Solution for Mathematics - Indefinite Integration: Evaluate $\int \frac{(x+1)}{x(1+x{e}^x{)}^2}dx$

Welcome, fellow traveler of the mathematical landscape. Today, we stand before an integral that, at first glance, might seem like a tangled mess of exponentials and polynomials:

It is easy to feel overwhelmed by the $(1+x{e}^x{)}^2$ in the denominator. But remember, in JEE Advanced, complexity is often just a mask for elegance. Let's peel back that mask together.

First, we must train our eyes to see the hidden relationships. Look at the term $x{e}^x$. What happens when we differentiate it?

Using the product rule, we get $e^x+x{e}^x$, which is $e^x(x+1)$. This is the 'spark' of our solution. We have $(x+1)$ in the numerator, but we are missing the $e^x$.

Forcing the Pattern

We don't just stare at the problem; we manipulate it. By multiplying the numerator and denominator by $e^x$, we create the perfect differential for our substitution.

Now, the integral becomes:

This is the turning point. By setting $t=1+x{e}^x$, we transform the entire expression into a rational function of $t$.

The numerator $e^x(x+1)dx$ becomes $dt$, and the denominator becomes $(t-1)t^2$. We have successfully moved from the world of exponentials to the world of algebra.

Breaking Down Barriers

Now, we use partial fraction decomposition. We break the expression into simpler components:

This is the beauty of calculus—taking a complex, intimidating structure and decomposing it into simple, solvable parts. Integrating these terms is straightforward:

Finally, we substitute back $t=1+x{e}^x$ to return to our original variable $x$.

The result is:

This is not just an answer; it is a testament to your ability to see through the noise. Keep practicing, keep questioning, and remember that every integral is a story waiting to be told.