Exercise 7.4 - Question 9
Problem
$$\int \frac{\sec^2 x}{\sqrt{\tan^2 x + 4}} \, dx$$
Step-by-Step Solution
Step 1: Identify the Substitution
Notice that sec²x is the derivative of tan x.
Let u = tan x, then du = sec²x dx
Step 2: Substitute
$$\int \frac{\sec^2 x}{\sqrt{\tan^2 x + 4}} \, dx = \int \frac{du}{\sqrt{u^2
+ 4}}$$
Step 3: Apply Standard Formula
Using the formula ∫1/√(u² + a²) du = ln|u + √(u² + a²)| + C with a = 2:
$$\int \frac{du}{\sqrt{u^2 + 4}} = \ln\left|u + \sqrt{u^2 + 4}\right| + C$$
Step 4: Substitute Back
$$= \ln\left|\tan x + \sqrt{\tan^2 x + 4}\right| + C$$
✅ Final Answer
$$\int \frac{\sec^2 x}{\sqrt{\tan^2 x + 4}} \, dx = \ln\left|\tan x +
\sqrt{\tan^2 x + 4}\right| + C$$