Exercise 7.4 - Question 20
Problem
$$\int \frac{x+2}{\sqrt{4x - x^2}} \, dx$$
Step-by-Step Solution
Step 1: Analyze
d/dx(4x - x²) = 4 - 2x = -2(x - 2)
Write x + 2 = A(-2x + 4) + B = -2Ax + 4A + B
Solving: -2A = 1, so A = -1/2; and 4A + B = 2, so B = 4
Step 2: Split
$$= -\frac{1}{2}\int \frac{4-2x}{\sqrt{4x-x^2}} \, dx + 4\int
\frac{1}{\sqrt{4x-x^2}} \, dx$$
Step 3: First Integral
$$I_1 = -\frac{1}{2} \cdot 2\sqrt{4x-x^2} = -\sqrt{4x-x^2}$$
Step 4: Second Integral
Complete the square: 4x - x² = 4 - (x-2)²
$$I_2 = 4\sin^{-1}\left(\frac{x-2}{2}\right)$$
✅ Final Answer
$$= -\sqrt{4x-x^2} + 4\sin^{-1}\left(\frac{x-2}{2}\right) + C$$