It seems you're referring to a process of solving an integral equation involving an integrating factor (i.f.), commonly used in the context of solving linear differential equations, such as in the method for solving first-order linear ODEs.

Here’s a breakdown of the steps you’re mentioning:

1. Find the integrating factor (i.f.): The integrating factor is typically of the form $\mu(x) = e^{\int P(x) \, dx}$, where $P(x)$ is the coefficient of $y(x)$ in the differential equation $\frac{dy}{dx} + P(x) y = Q(x)$.

2. Multiply through by the integrating factor: Once you've determined the integrating factor $\mu(x)$, you multiply the entire differential equation by $\mu(x)$, which transforms it into an equation that can be integrated more easily.

3. Substitute into the solution formula: After using the integrating factor, you use the formula: $y(x) \cdot \mu(x) = \int Q(x) \cdot \mu(x) \, dx + C$ where $C$ is the constant of integration, and $Q(x)$ is the term on the right-hand side of the original differential equation.

4. Solve for $y(x)$: Once you find the general solution to the equation, you divide both sides by $\mu(x)$ to solve for $y(x)$.

Example:

Let's go through a quick example.

Given a first-order linear ODE: $\frac{dy}{dx} + P(x) y = Q(x)$ The steps would be:

1. Find the integrating factor $\mu(x) = e^{\int P(x) \, dx}$.
2. Multiply the entire equation by $\mu(x)$.
3. The resulting equation becomes: $\mu(x) \frac{dy}{dx} + \mu(x) P(x) y = \mu(x) Q(x)$ or equivalently: $\frac{d}{dx} \left( y(x) \cdot \mu(x) \right) = \mu(x) Q(x)$
4. Integrate both sides: $y(x) \cdot \mu(x) = \int \mu(x) Q(x) \, dx + C$
5. Solve for $y(x)$: $y(x) = \frac{1}{\mu(x)} \left( \int \mu(x) Q(x) \, dx + C \right)$

Does this clarification help? Would you like to see a worked-out example of a specific problem or equation you're dealing with?