Certainly! Let's solve the integral

$\int \frac{2}{3 + \cos(2\theta)} \, d\theta$

without using the formula for $\int \frac{1}{a + b\cos(2\theta)} \, d\theta$.

Step 1: Use the double-angle identity

Recall that $\cos(2\theta) = 2\cos^2(\theta) - 1$. This allows us to express the integrand in terms of $\cos(\theta)$:

$\int \frac{2}{3 + \cos(2\theta)} \, d\theta = \int \frac{2}{3 + (2\cos^2\theta - 1)} \, d\theta = \int \frac{2}{2 + 2\cos^2\theta} \, d\theta.$

Simplifying, this becomes:

$\int \frac{1}{1 + \cos^2\theta} \, d\theta.$

Step 2: Use a trigonometric identity to simplify further

Now, recall the identity:

$\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}.$

Substituting this into the integral gives:

[ \int \frac{1}{1 + \frac{1 + \cos(2\theta)}{2}} , d\theta = \int \frac{2}{