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

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

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

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Key Concepts

Constant of Integration

The constant of integration represents the arbitrary constant added to an antiderivative when integrating. It accounts for all possible vertical shifts in the family of functions that share the same derivative, ensuring the general solution to the indefinite integral.

Antiderivative

An antiderivative is a function whose derivative is equal to the given function. In the context of indefinite integration, finding the antiderivative means determining a family of functions, each differing by a constant, that differentiates to the specified expression.

Substitution Method

The substitution method is a technique in integration used when an integrand is a composite function and one part of it is the derivative of another. This method, sometimes referred to as 'u-substitution', simplifies the integral by changing variables to reduce the expression to a more easily integrated form.

Chain Rule

The chain rule is a rule in differential calculus used to differentiate composite functions. Its reverse application is essential for integrating functions where the derivative of an inner function appears as a multiplicative factor, thus guiding the use of substitution in the integration process.

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