Question

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible. \(\int_{0}^{4 \pi} \cos (x / 2) \sin (x / 2) d x\)

Short answer

The integral evaluates to 0.

Step-by-step solution

Use Trigonometric Identity

Recognize that \(\cos(x/2)\sin(x/2)\) can be simplified using the trigonometric identity \(\cos(a)\sin(a) = \frac{1}{2}\sin(2a)\). In this case, \(a = x/2\), hence \(2a = x\). Thus, \(\cos(x/2)\sin(x/2) = \frac{1}{2}\sin(x)\). Substitute this identity into the integral.

Rewrite the Integral

Substitute the identity back into the integral:\[ \int_{0}^{4\pi}\cos(x/2)\sin(x/2)\, dx = \int_{0}^{4\pi}\frac{1}{2}\sin(x)\, dx = \frac{1}{2}\int_{0}^{4\pi}\sin(x)\, dx \]

Evaluate the Antiderivative

The antiderivative of \(\sin(x)\) is \(-\cos(x)\). Therefore, compute the antiderivative:\[ \frac{1}{2}\left[ -\cos(x) \right] \] evaluated from \(0\) to \(4\pi\).

Calculate the Definite Integral

Evaluate the definite integral from the antiderivative:\[ \frac{1}{2}\left( -\cos(4\pi) + \cos(0) \right) \]Since \(\cos(4\pi) = 1\) and \(\cos(0) = 1\), the expression becomes:\[ \frac{1}{2}\left( -1 + 1 \right) = \frac{1}{2}\times 0 = 0 \]

Conclusion

The value of the definite integral \(\int_{0}^{4\pi}\cos(x/2)\sin(x/2)\, dx\) is \(0\).

Key concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate different trigonometric functions to each other. These identities allow us to simplify complex trigonometric expressions and evaluate integrals or derivatives more easily. In the given problem, we start with the expression \(\cos(x/2)\sin(x/2)\). Here, we used the identity \(\cos(a)\sin(a) = \frac{1}{2}\sin(2a)\), helping us transform the expression to \(\frac{1}{2}\sin(x)\).
This simplification transforms the complex product of sine and cosine into a single sine function multiplied by a constant, making the integral calculation easier. This step is crucial as it breaks down the original problem into a form that is straightforward to integrate. Trigonometric identities like these often simplify problems in calculus and make solutions much more manageable. Understanding these identities and practicing their use is key for students to succeed in solving integration and differentiation problems effectively.

Antiderivatives

Antiderivatives, also known as indefinite integrals, are functions that reverse the process of differentiation. For any given function \(f(x)\), its antiderivative \(F(x)\) is such that \(F'(x) = f(x)\). In our exercise, the goal is to find the antiderivative of \(\sin(x)\), which in this context simplifies our integral.
The antiderivative of \(\sin(x)\) is \(-\cos(x)\). This means that differentiating \(-\cos(x)\) will give us \(\sin(x)\). Understanding these basic antiderivatives is a vital skill, since they allow us to evaluate integrals by calculating the area under curves of trigonometric functions. By rewriting the integral in terms of its antiderivative, we can easily find the value of a definite integral by substituting the limits of integration into \(-\cos(x)\). Knowing the common antiderivatives helps in quickly assessing such calculus problems.

Calculus Problems

Calculus problems often require a combination of different mathematical skills, such as algebraic manipulation, trigonometric identities, and the application of derivative and integral rules. In the given problem, we used elements from all these areas to find the solution.
Calculus problems typically involve solving expressions to find rates of change or to evaluate the total accumulation of quantities. In this case, we are dealing with a definite integral that gives us a precise numerical value, which is the accumulated value of the function from \(x = 0\) to \(x = 4\pi\). Solving calculus problems like this one involves several thoughtful steps: transforming the integrand using identities, finding the right antiderivative, then applying the limits of integration.
Mastering calculus problems requires practice and an understanding of when and how to apply these various mathematical tools and identities to find the most efficient and elegant solutions.