Exercise 7.4 - Question 15
Problem
$$\int \frac{1}{\sqrt{(x-a)(x-b)}} \, dx$$
Step-by-Step Solution
Step 1: Expand
$$(x-a)(x-b) = x^2 - (a+b)x + ab$$
Step 2: Complete the Square
$$= \left(x - \frac{a+b}{2}\right)^2 - \frac{(a+b)^2}{4} + ab = \left(x -
\frac{a+b}{2}\right)^2 - \frac{(a-b)^2}{4}$$
Step 3: Apply Standard Formula
This is of the form √(u² - c²) where u = x - (a+b)/2 and c = |a-b|/2:
$$\int \frac{1}{\sqrt{u^2 - c^2}} \, du = \ln\left|u + \sqrt{u^2 -
c^2}\right| + C$$
Step 4: Substitute Back
$$= \ln\left|x - \frac{a+b}{2} + \sqrt{(x-a)(x-b)}\right| + C$$
✅ Final Answer
$$\int \frac{1}{\sqrt{(x-a)(x-b)}} \, dx = \ln\left|x - \frac{a+b}{2} +
\sqrt{(x-a)(x-b)}\right| + C$$