Integrate the following function:

$\frac{1}{\sqrt{(x-1)(x-2)}}$

Question

Integrate the following function:

$\frac{1}{\sqrt{(x-1)(x-2)}}$

Accepted Answer

> [!TLDR]
> The key idea is to rewrite the integrand into a form matching $\int \frac{du}{\sqrt{u^2 - a^2}}$, which integrates to $\cosh^{-1}(u/a) + C$. After completing the square, the integral becomes $\cosh^{-1}(2x - 3) + C$.

## Why This Approach Works

When you see a square root of a quadratic in the denominator, your first instinct should be: **can I complete the square?** The expression $(x-1)(x-2)$ expands to $x^2 - 3x + 2$, which is a quadratic. The square root of a quadratic often hides a standard inverse hyperbolic or trigonometric form.

Specifically, the integral $\int \frac{dx}{\sqrt{x^2 - a^2}}$ is a known result: it equals $\cosh^{-1}(x/a) + C$ (or $\log|x + \sqrt{x^2 - a^2}| + C$, if you prefer the logarithmic form). Our job is to manipulate the given integrand until it matches that pattern.

The trick is to complete the square inside the root, then use a substitution that turns the expression into something of the form $\sqrt{u^2 - a^2}$.

---

## Step-by-Step Solution

**1. Expand and complete the square**

First, expand the product:
$$(x-1)(x-2) = x^2 - 3x + 2$$

Now complete the square for $x^2 - 3x$. Half of $-3$ is $-\frac{3}{2}$, so we add and subtract $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$:
$$x^2 - 3x + 2 = \left(x^2 - 3x + \frac{9}{4}\right) + 2 - \frac{9}{4} = \left(x - \frac{3}{2}\right)^2 - \frac{1}{4}$$

So the integral becomes:
$$\int \frac{dx}{\sqrt{\left(x - \frac{3}{2}\right)^2 - \frac{1}{4}}}$$

**2. Choose a substitution to simplify**

Let $u = x - \frac{3}{2}$. Then $du = dx$, and the integral is:
$$\int \frac{du}{\sqrt{u^2 - \frac{1}{4}}}$$

This is exactly the form $\int \frac{du}{\sqrt{u^2 - a^2}}$ with $a = \frac{1}{2}$.

> [!FORMULA]
> $$\int \frac{du}{\sqrt{u^2 - a^2}} = \cosh^{-1}\left(\frac{u}{a}\right) + C$$

**3. Apply the formula** …

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

Concept understanding — Integration By Completing Square

Integration by Completing the Square

The move

Case A — leads to inverse tangent

Case B — leads to a logarithm

Q.Integrate the following function: 1(x−1)(x−2)\frac{1}{\sqrt{(x-1)(x-2)}}(x−1)(x−2)​1​