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

Concept: Partial Fraction Decomposition — splitting a rational function into simpler fractions that integrate directly to logs.

Step 1: Set up the decomposition.

Since the denominator has distinct linear factors, write

Step 2: Solve for $A$ and $B$.

Multiply through by $(x+1)(x+2)$:

Comparing coefficients: $A + B = 1$ and $2A + B = 0$.

Subtracting gives $A = -1$, then $B = 2$.

Step 3: Integrate term by term.

The key idea here is Partial Fraction Decomposition. When you have a rational function (a polynomial divided by another polynomial) and the denominator factors into distinct linear factors, you can break the complicated fraction into a sum of simpler fractions — each with a single linear denominator. This turns a messy integration problem into a sum of easy logarithmic integrals.

Why does this work? The denominator $(x+1)(x+2)$ is already factored. The numerator $x$ is of lower degree than the denominator, so we can directly write:

where $A$ and $B$ are constants we need to find. Once we find them, integrating $\frac{A}{x+1}$ gives $A\log|x+1|$, and similarly for the other term.

Let’s find $A$ and $B$ step by step.

1. Set up the equation. Multiply both sides by the denominator $(x+1)(x+2)$ to clear fractions:

This identity must hold for all $x$.

2. Solve for $A$ and $B$. There are two efficient methods. I’ll use the substitution method (also called the cover-up method) because it’s fastest for distinct linear factors.
   • To find $A$, set $x = -1$ (this makes the $B$ term vanish because $x+1=0$):

• To find $B$, set $x = -2$ (this makes the $A$ term vanish):

3. Write the decomposed form. Substituting $A$ and $B$ back:

4. Integrate term by term. Now the integral becomes:

Each term is of the form $\int \frac{k}{ax+b} dx = \frac{k}{a} \log|ax+b| + C$. Here $a=1$ for both, so:

5. Simplify if desired. Using logarithm properties, $2\log|x+2| = \log(x+2)^2$, so we could also write:

But the form $-\log|x+1| + 2\log|x+2| + C$ is perfectly acceptable and often preferred.

Method: Partial fractions with distinct linear factors

Use this for a proper rational function whose denominator is a product of different linear factors, $\dfrac{N(x)}{(x-a)(x-b)\cdots}$. Each factor contributes one constant, and every piece integrates to a logarithm.

Steps

Step 1: Confirm the fraction is proper. Degree of numerator must be less than degree of denominator; if not, divide first.

Step 2: Write one constant per factor.

Step 3: Clear denominators and solve by the cover-up method — set $x$ equal to each root in turn so all but one term vanishes:

Step 4: Integrate term by term.

and sum the logs, keeping absolute values and the constant $C$.

Common Mistakes

Mistake 1: Dropping a constant's sign.

Why it's wrong: here $A=-1$, so the term is $-\log|x+1|$; carrying it as $+\log|x+1|$ flips the answer. Correct approach: solve $x=A(x+2)+B(x+1)$ carefully, and keep the sign the cover-up value produces.

Mistake 2: Omitting the absolute-value bars in the logs.

Why it's wrong: $\int\frac{dx}{x-a}=\log|x-a|$; without the bars the antiderivative is invalid where the argument is negative. Correct approach: always write $\log|x+1|$, $\log|x+2|$.

Mistake 3: Assuming the numerator constant equals the numerator of the original fraction.

Why it's wrong: for $\frac{x}{(x+1)(x+2)}$ the constants are $A=-1,B=2$, not $1$ — the numerator $x$ must be evaluated at each root. Correct approach: use the cover-up value $A=\frac{x}{x+2}\big|_{x=-1}$, etc.