For questions about integrating factors in general as well as their application to solving ODEs.
Integrating factors are commonly used to solve ordinary differential equations, oftentimes first order linear ODEs, though their use is not strictly limited to this class of problems. In the case of a first order linear ODE, $$\frac{dy}{dx} +p(x)y =q(x),\tag{1}$$ an integrating factor can be obtained as follows. Let $\mu(x)$ be a differentiable function to be determined later and multiply $(1)$ by $\mu(x)$: $$\mu(x)\frac{dy}{dx} + \mu(x)p(x)y = \mu(x)q(x).\tag{2}$$ Observe that $$\frac{d}{dx}\left(\mu(x)y\right) = \mu(x)\frac{dy}{dx} + \frac{d\mu}{dx}y.\tag{3}$$ By noting the similarity in the LHS of $(2)$ and the RHS of $(3)$ we choose $\mu(x)$ to solve $$\frac{d\mu}{dx} = \mu(x)p(x),$$ that is $$\mu(x) = e^{\int_{s_{0}}^{x}p(s)\,ds}.$$ This $\mu$ is then our integrating factor, and we can rewrite $(2)$ as $$\frac{d}{dx}\left(y e^{\int_{s_{0}}^{x}p(s)\,ds}\right) = q(x)e^{\int_{s_{0}}^{x}p(s)\,ds}.$$
