Example of Integration.

Integrate ∫ [log₄(x²) + log₄(3x)] .

The integral of the given expression can be written as:

∫ [log₄(x²) + log₄(3x)] dx

Using the property of logarithms that states log₄(a) + log₄(b) = log₄(a × b), we can rewrite the expression as:

∫ log₄(x² × 3x) dx

Now, we can use the property of integrals that states:

 ∫ log(f(x)) dx = x × log(f(x)) - ∫ x / f(x) df(x).

Applying this property, we get:

∫ log₄(x² × 3x) dx = x × log₄(x² × 3x) - ∫ x / (x² × 3x) d(x² × 3x)

Simplifying the expression, we get:

∫ log₄(x² × 3x) dx = x × log₄(x² × 3x) - ∫ 1 / (3x²) dx

Integrating the remaining term, we get:

∫ log₄(x² × 3x) dx = x × log₄(x² × 3x) - (1 / 3) × x / x² + C

where C is the constant of integration.

Therefore, the integral of the given expression is:

∫ [log₄(x²) + log₄(3x)] dx = x × log₄(x² × 3x) - (1 / 3) × x / x² + C