Exercise 7.4 - Question 21
Problem
$$\int \frac{x+2}{\sqrt{x^2 + 2x + 3}} \, dx$$
Step-by-Step Solution
Step 1: Express Numerator
d/dx(x² + 2x + 3) = 2x + 2
Write x + 2 = A(2x + 2) + B = 2Ax + 2A + B
Solving: 2A = 1, so A = 1/2; and 2A + B = 2, so B = 1
Step 2: Split
$$= \frac{1}{2}\int \frac{2x+2}{\sqrt{x^2+2x+3}} \, dx + \int
\frac{1}{\sqrt{x^2+2x+3}} \, dx$$
Step 3: First Integral
$$I_1 = \frac{1}{2} \cdot 2\sqrt{x^2+2x+3} = \sqrt{x^2+2x+3}$$
Step 4: Second Integral
Complete the square: x² + 2x + 3 = (x+1)² + 2
$$I_2 = \ln\left|(x+1) + \sqrt{x^2+2x+3}\right|$$
✅ Final Answer
$$= \sqrt{x^2+2x+3} + \ln\left|(x+1) + \sqrt{x^2+2x+3}\right| + C$$