Choosing an Antiderivative In Exercises 1-4, select the...

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative. $\frac{d y}{d x}=\frac{x}{\sqrt{x^{2}+1}}$ $$ \begin{array}{ll}{\text { (a) } 2 \sqrt{x^{2}+1}+C} & {\text { (b) } \sqrt{x^{2}+1}+C} \\ {\text { (c) } \frac{1}{2} \sqrt{x^{2}+1}+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array} $$

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative.
$\frac{d y}{d x}=\frac{x}{\sqrt{x^{2}+1}}$
$$
\begin{array}{ll}{\text { (a) } 2 \sqrt{x^{2}+1}+C} & {\text { (b) } \sqrt{x^{2}+1}+C} \\ {\text { (c) } \frac{1}{2} \sqrt{x^{2}+1}+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array}
$$

Key Concepts

Integration by Substitution

Integration by substitution is a method used to simplify an integration problem by replacing a part of the integrand with a new variable. This technique is particularly useful when the derivative of a chosen substitution is present elsewhere in the integrand, allowing the integral to be transformed into an easier form.

Chain Rule

The chain rule is a differentiation technique used to differentiate composite functions. In the context of verifying antiderivatives, the chain rule helps determine if a candidate function, when differentiated, correctly produces the original function by accounting for the derivative of the inner function.

Antiderivative

An antiderivative is a function whose derivative equals the original function. It represents the inverse process of differentiation and is fundamental in integration, where finding the antiderivative of a function leads to its indefinite integral, including an arbitrary constant.

Indefinite Integration

Indefinite integration is the process of finding a function (or family of functions) whose derivative is the given integrand. Since differentiation loses constant terms, the result of an indefinite integration includes an arbitrary constant of integration.

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