Exercise 7.4 - Question 13
Problem
$$\int \frac{1}{\sqrt{(x-1)(x-2)}} \, dx$$
Step-by-Step Solution
Step 1: Expand the Product
$$(x-1)(x-2) = x^2 - 3x + 2$$
Step 2: Complete the Square
$$x^2 - 3x + 2 = \left(x - \frac{3}{2}\right)^2 - \frac{9}{4} + 2 = \left(x
- \frac{3}{2}\right)^2 - \frac{1}{4}$$
Step 3: Rewrite the Integral
$$\int \frac{1}{\sqrt{(x - 3/2)^2 - (1/2)^2}} \, dx$$
Step 4: Apply Standard Formula
Using ∫1/√(u² - a²) du = ln|u + √(u² - a²)| with u = x - 3/2, a = 1/2:
$$= \ln\left|x - \frac{3}{2} + \sqrt{(x-\frac{3}{2})^2 - \frac{1}{4}}\right|
+ C$$
$$= \ln\left|x - \frac{3}{2} + \sqrt{(x-1)(x-2)}\right| + C$$
✅ Final Answer
$$\int \frac{1}{\sqrt{(x-1)(x-2)}} \, dx = \ln\left|x - \frac{3}{2} +
\sqrt{(x-1)(x-2)}\right| + C$$