The integral of f(x) * g(x) dx does not have a general, simple formula like the integral of a sum or a constant multiple. Instead, it is usually solved using a technique called integration by parts.

Integration by parts comes from the product rule for differentiation. The formula is:

Where:

• u = f(x) or g(x) (a function to be differentiated)
• dv = the remaining part of the integrand, including dx (a function to be integrated)
• du = derivative of u
• v = integral of dv

Steps to apply integration by parts:

1. Choose u and dv from the integrand f(x)g(x) dx. A helpful guideline is the acronym LIATE:
   • Logarithmic functions
   • Inverse trigonometric functions
   • Algebraic functions
   • Trigonometric functions
   • Exponential functions
   Choose u based on what comes first in LIATE. The rest becomes dv.
2. Calculate du (the derivative of u) and v (the integral of dv).
3. Apply the integration by parts formula: ∫ u dv = uv - ∫ v du
4. Evaluate the new integral ∫ v du. If it's simpler than the original, the choice of u and dv was good. If not, you might need to try a different choice for u and dv or apply integration by parts again.

Example:

Evaluate ∫ x cos(x) dx

1. Let u = x (algebraic) and dv = cos(x) dx (trigonometric).
2. Then du = dx and v = ∫ cos(x) dx = sin(x).
3. Apply the formula: ∫ x cos(x) dx = x sin(x) - ∫ sin(x) dx
4. Evaluate the new integral: ∫ sin(x) dx = -cos(x)
5. Therefore, ∫ x cos(x) dx = x sin(x) - (-cos(x)) + C = x sin(x) + cos(x) + C, where C is the constant of integration.

In summary, there is no direct general formula for the integral of f(x) * g(x) dx. You typically need to use integration by parts.