Substitution in Definite Integrals

Definition

Substitution in Definite Integrals is an extension of the $u$ -substitution method that includes the transformation of the limits of integration. This allows for the evaluation of a definite integral entirely in terms of the new variable.

Why It Matters

Definite substitution is a “morphism of intervals” that significantly reduces algebraic error; by transforming the limits of integration along with the variable, we can evaluate a problem entirely in a new space, eliminating the risky and time-consuming step of back-substitution.

Core Concepts

Limit Transformation: When changing $x$ $x$ to $u$ $u$ using $u = g(x)$ $u = g (x)$ , the $x$ $x$ -limits $a$ $a$ and $b$ $b$ must be converted to $u$ $u$ -limits $g(a)$ $g (a)$ and $g(b)$ $g (b)$ .
- How to read: “u equals g of x; x-limits a and b become u-limits g(a) and g(b).”
- Meaning: The substitution maps the interval $[a,b]$ to $[g(a), g(b)]$ . Transform limits before integrating.
Formula: $\int_a^b f(g(x)) g'(x) dx = \int_{g(a)}^{g(b)} f(u) du$ $\int_{a}^{b} f (g (x)) g^{'} (x) d x = \int_{g (a)}^{g (b)} f (u) d u$ .
- How to read: “Integral from a to b of f of g of x times g-prime of x dx equals integral from g(a) to g(b) of f of u du.”
- Meaning / when to use: When $u = g(x)$ , both the integrand and the limits transform. Evaluate entirely in $u$ -space—no back-substitution needed.
No Back-Substitution: Once the limits are converted, the problem is completely redefined in terms of $u$ ; there is no need to return to the variable $x$ .

Substitution in Definite Integrals

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes