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