Animated Solution for Mathematics - Indefinite Integration: Evaluate $\int \frac{(x-1)e^x}{(x+1{)}^3}dx$

Welcome, fellow traveler of the mathematical landscape. Today, we are going to unravel a problem that, at first glance, might seem like a daunting task of integration by parts.

We are looking at the integral:

When you see an exponential function $e^x$ multiplied by a rational function, your intuition should immediately shift toward a specific, beautiful identity:

This identity is a bridge that turns a complex integral into a single-step triumph.

The challenge here is that our integrand does not immediately show us $f(x)$ and $f^{\prime }(x)$. We have a single fraction $\frac{x-1}{(x+1{)}^3}$.

To unlock the identity, we must force the numerator to resemble the denominator. We look at the denominator, $(x+1{)}^3$, and realize that if we can create an $(x+1)$ in the numerator, we can simplify the fraction.

So, we perform a simple yet powerful algebraic trick: rewrite $(x-1)$ as $(x+1)-2$. Now, our integral becomes:

By distributing the denominator, we split the fraction into two distinct parts:

Now, look closely. Let us test the hypothesis that $f(x)=\frac{1}{(x+1{)}^2}$.

If we write this as $f(x)=(x+1{)}^{-2}$, we can easily find its derivative using the power rule. Differentiating $f(x)$ gives us:

It is a perfect match! The second term in our integral is indeed the derivative of the first.

With the structure $f(x)+f^{\prime }(x)$ confirmed, the integral collapses into the elegant form $e^{x}f(x)+C$.

Substituting our $f(x)$ back in, we arrive at the final result:

This problem teaches us that in mathematics, as in life, sometimes the most complex-looking obstacles are simply waiting for the right perspective to reveal their underlying simplicity. Keep practicing, keep questioning, and keep falling in love with the logic behind the math.