Exercise 7.4 - Question 19
Problem
$$\int \frac{6x+7}{\sqrt{(x-5)(x-4)}} \, dx$$
Step-by-Step Solution
Step 1: Expand Denominator
$$(x-5)(x-4) = x^2 - 9x + 20$$
Derivative: 2x - 9
Write 6x + 7 = A(2x - 9) + B, solving: A = 3, B = 34
Step 2: Split
$$= 3\int \frac{2x-9}{\sqrt{x^2-9x+20}} \, dx + 34\int
\frac{1}{\sqrt{x^2-9x+20}} \, dx$$
Step 3: First Integral
$$I_1 = 3 \cdot 2\sqrt{x^2-9x+20} = 6\sqrt{(x-5)(x-4)}$$
Step 4: Second Integral
Complete the square: x² - 9x + 20 = (x - 9/2)² - 1/4
$$I_2 = 34\ln\left|x - \frac{9}{2} + \sqrt{(x-5)(x-4)}\right|$$
✅ Final Answer
$$= 6\sqrt{(x-5)(x-4)} + 34\ln\left|x - \frac{9}{2} +
\sqrt{(x-5)(x-4)}\right| + C$$