# Integration Using U-Substitution The process of evaluating an integral is we must first find the antiderivative of a function. - Although in the case of definite integrals, the The process of integrating a function is not always easy, integration by substitution helps with this by combining basic concepts of function substitution and the chain rule. While there is no direct “chain rule of integration”, however, the technique of $u$-substitution is somewhat analagous to it. The substitution ## Choosing The Value of U ## Substitution of Indefinite Integrals > [!info]- Substitution with Indefinite Integrals > 1. Let $u=g(x)$, where $g′(x)$ is continuous over an interval > 2. Let $f(x)$ be continuous over the corresponding range of the function $g$ > 3. Let $F(x)$ be an antiderivative of $f(x)$ > > Then: > $\int f(g(x))g'(x)dx = \int f(u)du$ > $= F(u) + C$ > $= F(g(x)) + C$ ## Substitution of Definite Integrals Using substitution with definite integrals requires changing the limits of integration ($\int^a_b$)l > [!info]- Substitution with Definite Integrals > 1. Let $u = g(x)$ > 2. Let $g'$ be continuous over an interval $[a,b]$ > 3. Let $f$ be continuous over the range of $u=g(x)$ > > Then: > $\int^{b}_{a} f(g(x)) \cdot g'(x) \cdot dx = \int^{g(b)}_{g(a)} f(u)du$ ## Determining What To Substitute The following guidelines can be followed when determining what oo of the equation to replace with $u$: 1. At least one of the original variable must remain in the integrand after substituting $u$. 2. The derivative of $u$, or $g'(x)$, must cancel out what remains in the original function. [Integration by Parts (How to Choose U)](https://www.youtube.com/watch?v=XKmRIafsbGw) - when the variable is in the denominator