The composite function ƒ(g(𝓍)) consists of an inner function g and an outer function ƒ. If an integrand includes ƒ(g(𝓍)), which function is often a likely choice for a new variable u?

Verified step by step guidance

1. Identify the composite function given as ƒ(g(𝓍)), where g(𝓍) is the inner function and ƒ is the outer function.
2. Recall that when performing integration involving composite functions, substitution is a common technique to simplify the integral.
3. In substitution, we typically choose the inner function g(𝓍) as the new variable u because it simplifies the integrand and its differential du relates directly to dx.
4. Express the substitution as u = g(𝓍), then compute the differential du = g\' (𝓍) d𝓍 to replace parts of the integral accordingly.
5. Rewrite the integral entirely in terms of u and du, which often makes the integral easier to evaluate.

Key Concepts

Composite Functions

A composite function is formed when one function is applied to the result of another, written as ƒ(g(x)). Understanding how the inner function g(x) and outer function ƒ relate is essential for manipulating and simplifying expressions involving compositions.

U-Substitution Method

U-substitution is a technique used in integration where a new variable u is chosen to simplify the integral. Typically, u is set equal to the inner function g(x) in a composite function to make the integral easier to evaluate.

Chain Rule in Integration

The chain rule relates the derivative of a composite function to the derivatives of its inner and outer functions. In integration, recognizing this structure helps identify the inner function as a candidate for substitution, reversing the chain rule process.