**Integration of dx/(cos^2x * √3sin^2x)**

To integrate the given expression, dx/(cos^2x * √3sin^2x), we can use trigonometric identities and substitution method. Let's break down the solution into steps for better understanding.

**Step 1: Simplify the expression**

First, let's simplify the expression by using trigonometric identities. We know that cos^2x = 1 - sin^2x, so we can rewrite the expression as:

dx/((1 - sin^2x) * √3sin^2x)

**Step 2: Apply substitution**

To proceed with the integration, let's make a substitution. Let's substitute sinx with t, which means x = arcsin(t). Also, dx = dt/√(1 - t^2).

Now, the expression becomes:

(dt/√(1 - t^2)) / ((1 - t^2) * √3t^2)

Simplifying further:

dt / (√(1 - t^2) * (1 - t^2) * √3t^2)

**Step 3: Separate the terms**

Let's separate the terms in the denominator:

dt / (√(1 - t^2) * (1 - t^2) * √(3) * t^2)

**Step 4: Simplify the expression**

To simplify the expression further, we can cancel out common terms from the numerator and denominator:

dt / (√(1 - t^2) * (1 - t^2) * √(3) * t^2)

= dt / (√3 * (1 - t^2)^(3/2) * t^2)

**Step 5: Integrate the expression**

Now, we can integrate the expression after simplification. The integral becomes:

∫ dt / (√3 * (1 - t^2)^(3/2) * t^2)

**Step 6: Apply integration formula**

To integrate the above expression, we can use the formula:

∫ dx / (a^2 - x^2)^n = (1/(a^2(n-1))) * (x/(a^2 - x^2)^(n-1)) + (2n-3)/(a^2(n-1)) * ∫ dx / (a^2 - x^2)^(n-1)

Applying the formula to our integral:

∫ dt / (√3 * (1 - t^2)^(3/2) * t^2) = (1/(√3 * 2 * (3/2))) * (t/((1 - t^2)^(3/2))) + (2 * (3/2) - 3)/(√3 * 2 * (3/2)) * ∫ dt / ((1 - t^2)^(3/2))

Simplifying further:

= (1/(2√3)) * (t/((1 - t^2)^(3/2))) + (3 - 3)/(2√3) * ∫ dt / ((1 - t^2)^(3/2))

= (1/(2√3)) * (t