91Ó°ÊÓ

Trigonometric functions are fundamental in calculus, particularly when dealing with periodic functions such as sine and cosine. In our exercise, the function includes \( \cos^2 x \), a common trigonometric function that often requires simplification to integrate effectively.
To work with trigonometric integrals, we frequently use identities that simplify the functions involved. For instance, the power-reduction identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) is powerful.
It transforms a squared trigonometric function into a form that is easier to integrate. This identity allows us to rewrite \( \cos^2 x \) into expressions of half-angles which results in simpler integrands.
Understanding and applying trigonometric identities helps in efficiently breaking down complex functions into simpler parts, setting the stage for successful integration.

Mastering integration techniques is essential for solving a wide array of calculus problems, including finding antiderivatives of complex functions. For the expression \( f(x) = \frac{3}{2} + \frac{1}{2} \cos(2x) \), we encounter a straightforward process due to its simpler structure.
When integrating, two main techniques are:

• Direct Integration: If the function is a sum of simple terms like a constant or basic trigonometric expression, it can be integrated term by term.
• Using Identities: Trigonometric identities, like in this problem, assist in breaking down expressions into more integrable forms.

Each term of the given function can be dealt with separately.
The antiderivative of a constant \( \frac{3}{2} \) is simply \( \frac{3}{2}x \), while for \( \frac{1}{2}\cos(2x) \), we rely on the substitution method to handle the cosine with a linear argument.

The substitution method is a powerful technique for integrating functions where the integrand involves a composite function. It simplifies this process by converting the integral of a complex function into an easier, standard form.
In our exercise, this method plays a crucial role when integrating the term \( \frac{1}{2} \cos(2x) \).
Substitution is made by recognizing that the inner function, \( 2x \), requires us to compensate by dividing by the derivative of \( 2x \), which is 2. Hence the integral of \( \cos(2x) \) becomes \( \frac{1}{2} \sin(2x) \), and with the additional coefficient, the term is fully integrated to \( \frac{1}{4}\sin(2x) \).
The key steps in substitution include:

• Identifying the inner function and its derivative.
• Adjusting the integral accordingly by compensating for the derivative.

This method not only aids in quicker calculations but also in understanding the underlying structure of composite functions in calculus.