Exercise 7.4 - Question 23
Problem
$$\int \frac{5x+3}{\sqrt{x^2 + 4x + 10}} \, dx$$
Step-by-Step Solution
Step 1: Express Numerator
d/dx(x² + 4x + 10) = 2x + 4
Write 5x + 3 = A(2x + 4) + B
Solving: 2A = 5, so A = 5/2; and 4A + B = 3, so B = -7
Step 2: Split
$$= \frac{5}{2}\int \frac{2x+4}{\sqrt{x^2+4x+10}} \, dx - 7\int
\frac{1}{\sqrt{x^2+4x+10}} \, dx$$
Step 3: First Integral
$$I_1 = \frac{5}{2} \cdot 2\sqrt{x^2+4x+10} = 5\sqrt{x^2+4x+10}$$
Step 4: Second Integral
Complete the square: x² + 4x + 10 = (x+2)² + 6
$$I_2 = 7\ln\left|(x+2) + \sqrt{x^2+4x+10}\right|$$
✅ Final Answer
$$= 5\sqrt{x^2+4x+10} - 7\ln\left|(x+2) + \sqrt{x^2+4x+10}\right| + C$$