Q.Integrate the following function: $\frac{x e^x}{(1+x)^2}$

The key idea is to rewrite the integrand so that the Power Rule for integration (or a standard form) applies, often after noticing a derivative relationship.

We have:

Step 1: Write the numerator as $x e^x = (x+1-1)e^x = (x+1)e^x - e^x$.

Step 2: Split the fraction:

$\int \frac{(x+1)e^x}{(1+x)^2} \, dx - \int \frac{e^x}{(1+x)^2} \, dx = \int \frac{e^x}{1+x} \, dx - \int \frac{e^x}{(1+x)^2} \, dx$ …

We are integrating $\frac{x e^x}{(1+x)^2}$. At first glance, this looks like a candidate for integration by parts, but there is a more elegant approach. Notice the denominator $(1+x)^2$ and the numerator $x e^x$. The presence of $e^x$ and a polynomial suggests that the derivative of something like $\frac{e^x}{1+x}$ might appear.

Let’s check: differentiate $\frac{e^x}{1+x}$ using the quotient rule:

That is exactly our integrand! So the integral is simply the antiderivative we just found.

Now, let’s work through it step by step to see why this works and how you might spot it yourself.

1. Recognize the pattern: The integrand has a denominator $(1+x)^2$ and a numerator with $e^x$ times $x$. When you see $e^x$ multiplied by a rational function, think about the derivative of $\frac{e^x}{\text{something}}$. The derivative of $e^x f(x)$ is $e^x(f(x) + f'(x))$, but here the denominator is squared, hinting at a quotient rule structure.

2. Guess a candidate: Try $F(x) = \frac{e^x}{1+x}$. Compute its derivative:

This matches perfectly. So the antiderivative is $F(x) + C$.

3. Verify by differentiation: If you are ever unsure, differentiate your answer. Here,

confirming correctness. …

Method: The $e^{x}\big(f(x)+f'(x)\big)$ shortcut

Whenever an integrand is $e^{x}$ times a bracket that is a function plus its own derivative, the answer is simply $e^{x}$ times that function:

Steps

Step 1: Isolate the $e^{x}$ factor and look at what multiplies it.

Step 2: Try to split the multiplier as $f(x)+f'(x)$.

Guess a candidate $f(x)$ (often the "nicer" of the two pieces, or the part left after a $\tfrac1x$-type term), then check that its derivative supplies the remaining piece. …

Common Mistakes

Mistake 1: Not recognising the $e^{x}(f+f')$ pattern hidden by the algebra.

Why it's wrong: $\dfrac{x}{(1+x)^{2}}=\dfrac{1}{1+x}-\dfrac{1}{(1+x)^{2}}$, which is $f+f'$ with $f=\dfrac{1}{1+x}$. Correct approach: rewrite before integrating.

Mistake 2: Getting the sign of $f'(x)$ wrong. …