Exercise 7.4 - Question 17
Problem
$$\int \frac{x+2}{\sqrt{x^2 - 1}} \, dx$$
Step-by-Step Solution
Step 1: Split the Integral
$$\int \frac{x+2}{\sqrt{x^2 - 1}} \, dx = \int \frac{x}{\sqrt{x^2 - 1}} \,
dx + 2\int \frac{1}{\sqrt{x^2 - 1}} \, dx$$
Step 2: First Integral (I₁)
For ∫x/√(x²-1) dx, let u = x² - 1, du = 2x dx:
$$I_1 = \frac{1}{2}\int \frac{du}{\sqrt{u}} = \sqrt{u} = \sqrt{x^2 - 1}$$
Step 3: Second Integral (I₂)
$$I_2 = 2\int \frac{1}{\sqrt{x^2 - 1}} \, dx = 2\ln\left|x + \sqrt{x^2 -
1}\right|$$
Step 4: Combine
$$= \sqrt{x^2 - 1} + 2\ln\left|x + \sqrt{x^2 - 1}\right| + C$$
✅ Final Answer
$$\int \frac{x+2}{\sqrt{x^2 - 1}} \, dx = \sqrt{x^2 - 1} + 2\ln\left|x +
\sqrt{x^2 - 1}\right| + C$$