Q.Find $\int \dfrac{dx}{(x+1)(x+2)}$

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

Step 1: Decompose

Write

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

Step 2: Solve for constants

Set $x = -1$: $1 = A(1) \Rightarrow A = 1$.

Set $x = -2$: $1 = B(-1) \Rightarrow B = -1$.

Step 3: Integrate

The key idea here is that the integrand $\frac{1}{(x+1)(x+2)}$ is a rational function whose denominator factors into two distinct linear factors. When you have such a product in the denominator, you can break the fraction into a sum of two simpler fractions — each with one of the linear factors in the denominator. This is called Partial Fraction Decomposition.

Why does this help? Because integrating $\frac{1}{x+1}$ or $\frac{1}{x+2}$ is immediate — each gives a natural logarithm. But integrating the original product form directly is not obvious. So we rewrite the problem into something we already know how to handle.

Let’s work through it step by step.

1. Set up the decomposition. We want constants $A$ and $B$ such that:

This equality must hold for all $x$ (except where denominators vanish).

2. Clear the denominators. Multiply both sides by $(x+1)(x+2)$:

This is an identity in $x$.

3. Solve for $A$ and $B$. Expand the right-hand side:

For this to hold for all $x$, the coefficients of $x$ and the constant term must match on both sides. So:

From the first equation, $B = -A$. Substitute into the second:

Then $B = -1$.

4. Rewrite the integral. Now we have:

5. Integrate term by term. Each integral is a standard form:

Combining constants:

6. Simplify using logarithm properties. The difference of logs is the log of a quotient:

So the final antiderivative is:

Method: Partial Fractions for Distinct Linear Factors

Use this when the integrand is a proper fraction whose denominator factors into distinct linear terms: split it into simple fractions, each integrating to a logarithm.

Steps

Step 1: Write the decomposition.

For $\dfrac{1}{(x+a)(x+b)}$ set

Step 2: Solve for the constants.

Clear denominators: $1=A(x+b)+B(x+a)$. Substitute the roots $x=-a$ and $x=-b$ to read off $A$ and $B$ quickly.

Step 3: Integrate each simple fraction.

Each term gives a log: $\int\frac{A}{x+a}\,dx=A\log|x+a|$. Combine, using log laws where the coefficients allow, and add $C$. For $(x+1)(x+2)$ this yields $\log\left|\dfrac{x+1}{x+2}\right|+C$.

Common Mistakes

Mistake 1: Applying partial fractions to an improper fraction.

Why it's wrong: the method needs numerator degree < denominator degree; otherwise divide first. Correct approach: check the degrees before decomposing (here it is proper, so proceed).

Mistake 2: Sign errors solving for $A$ and $B$.

Why it's wrong: substituting the wrong root or sign gives $A$ and $B$ swapped, flipping the log argument. Correct approach: substitute $x=-1$ and $x=-2$ carefully.

Mistake 3: Forgetting the modulus in the logarithm.

Why it's wrong: $\int\frac{1}{x+1}\,dx=\log|x+1|$; without the modulus the answer is invalid for negative arguments. Correct approach: always write $\log|\cdot|$.