Finding u and d u In Exercises 1-4, complete the table by...

Finding $u$ and $d u$ In Exercises $1-4,$ complete the table by identifying $u$ and $d u$ for the integral.
$$
\begin{array}{l}{\int f(g(x)) g^{\prime}(x) d x \quad u=g(x) \quad d u=g^{\prime}(x) d x} \\ {\int\left(8 x^{2}+1\right)^{2}(16 x) d x}\end{array}
$$

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Key Concepts

Integration by Substitution

This is a technique used in calculus to simplify the process of integration by making a change of variable. When confronted with an integral that involves a composite function, substitution allows one to rewrite the integral in a simpler form by introducing a new variable, typically denoted as u, which absorbs part of the composite function and its differential, thereby reducing the integral to a standard form.

Chain Rule

The chain rule is a fundamental concept in differential calculus that allows for the differentiation of composite functions. In the context of u-substitution, the chain rule explains how the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This relationship is key to identifying the correct differential du when performing the substitution.

Differentials

Differentials, denoted as du or dx, represent an infinitesimally small change in the variable. When performing u-substitution, once you define u as a function of x, you then differentiate u to find du. Matching parts of the original integral with du is essential for transforming the integral from the variable x to the variable u, thus facilitating its evaluation.

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