Find \(\int \frac{x+\cos 2 x}{3} \mathrm{~d} x\).

The concept of an antiderivative is fundamental in integral calculus. It refers to the reverse process of differentiation. For any given function, an antiderivative is another function whose derivative is the original function. This is akin to finding the original shape of a piece of clay from its mold.

Consider a function represented as \(f(x)\), its antiderivative, often denoted as \(F(x)\), fulfills the condition that \( F'(x) = f(x) \). Since the derivative of a constant value is zero, antiderivatives are not unique; they belong to a family of functions differing by a constant, represented as \( + C \), where \(C\) is the constant of integration. In the context of the problem \( \int \frac{x + \cos 2x}{3} \, dx \), finding the antiderivative of each term provides us with the building blocks to solve the integral.

The power rule for integration is a quick method to find the antiderivative of a function of the form \( x^n \), where \( n \) is any real number excluding \(-1\). The rule states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), plus the constant of integration.

In our example, we apply the power rule to the function \( \frac{x}{3} \). Using the power rule, we integrate \( x^1 \) to obtain \( \frac{x^2}{2} \), and then we factor out the coefficient \( \frac{1}{3} \) from the integral to simplify our expression to \( \frac{x^2}{6} + C_1 \). This represents the power rule in action, providing an efficient path to the solution.

Integrating trigonometric functions is another vital aspect of integral calculus. Some of the basic trigonometric integrals include the integrals of \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \), among others. Each trigonometric function has its antiderivative that can be memorized or derived from known identities.

In our problem, the function to integrate is \( \cos(2x) \), divided by 3. To integrate this function, you would normally find the antiderivative of \( \cos(x) \), which is \( \sin(x) \), and then adjust the result by the outside factor and the argument inside the trigonometric function. Therefore, the integral \( \frac{1}{3} \int \cos(2x) \, dx \) becomes \( \frac{\sin(2x)}{6} + C_2 \). This demonstrates the application of trigonometric integration.

Integration by substitution, sometimes referred to as the reverse chain rule, is a method used to simplify integrals. When faced with a complex integral, substitution allows us to replace a portion of the integral with a new variable, making it easier to integrate. This technique often involves recognizing a part of the integrand function that can be considered as \( u'(x) \) for some function \( u(x) \), and then rewriting the integral in terms of \( u \).

In the example given, we have the integral of \( \frac{\cos 2x}{3} \). By letting \( u = 2x \), we find that \( dx = \frac{1}{2} du \). Substituting these values into the integral simplifies it to \( \frac{1}{6} \int \cos u \, du \). After integrating, we return to the original variable, resulting in the antiderivative \( \frac{\sin 2x}{6} + C_2 \). This illustrates the effectiveness of integration by substitution for transforming and solving integrals.