Problem 49 - Calculus: One and Several Variables

Question

Calculate using our table of integrals. $$\begin{aligned} &\text { Evaluate } \int_{0}^{ \pi} \sqrt{1+\cos x} d x\\\ &\mathrm{HINT}: \cos x=2 \cos ^{2} \frac{5}{2} x-1 \end{aligned}$$

Step-by-Step Solution

Answer

The short answer is: \(\int_{0}^{\pi}\sqrt{1+\cos x} dx = 4\).

Step 1: Identify and apply the HINT

Given the HINT, we know that \(\cos x=2 \cos^2 \frac{x}{2}-1\). Our integral expression can be written using the HINT as: \[ \int_{0}^{\pi}\sqrt{1+\left(2\cos^2\frac{x}{2} -1\right)} dx \]

Step 2: Simplify the integral expression

We can then simplify the expression inside the square root as: \[ \begin{aligned} &\int_{0}^{\pi}\sqrt{2 \cos^2 \frac{x}{2}} dx \\ \Rightarrow\ &2\int_{0}^{\pi}\lvert\cos\frac{x}{2}\rvert dx \end{aligned} \]

Step 3: Integrate using the table of integrals/use periodic identity

We can now calculate the integral of the simplified expression. Since \(\cos x\) is an even function and will always be positive on the interval \([0, \pi]\), the absolute value of \(\cos(\frac{x}{2})\) is taken on that interval: \[ \begin{aligned} 2\int_{0}^{\pi}\cos\frac{x}{2} dx \end{aligned} \] To integrate, we use the antiderivative of the cosine function, which is the sine function: \[ \begin{aligned} \left[2\cdot \left(2\sin\frac{x}{2}\right)\right]_0^{\pi} \end{aligned} \]

Step 4: Determine the result with the given limits

Finally, we use the given limits, \(x = 0\) and \(x = \pi\), to evaluate the definite integral: \[ \begin{aligned} 2(2 \sin(\frac{\pi}{2}) - 2 \sin(0)) = 4 \end{aligned} \] Thus, the value of the integral is: \[ \begin{aligned} \int_{0}^{\pi}\sqrt{1+\cos x} dx = 4 \end{aligned} \]

Key Concepts

Table of Integrals

A Table of Integrals is an essential tool for solving integrals that are too complex to handle manually. It lists a variety of integrals and their antiderivatives, making it easier to find solutions without performing lengthy calculations. In mathematics, especially calculus, integrations can get tricky, and memorizing every possible form of an integral is impractical.

• It provides a quick reference to known integral solutions, saving time.
• Formulas in the table include integrals for trigonometric, exponential, and logarithmic functions, among others.

In our exercise, the table of integrals helps us handle the integration of the \( \sqrt{1+\cos x} \) term once we have rewritten it using a trigonometric identity. By referencing the table, we avoid lengthy algebraic manipulations, focusing instead on simplifying preliminary expressions using identities and properties of integrals.

Trigonometric Identities

Trigonometric Identities are equations involving trigonometric functions that hold true for all input values. They are essential in simplifying expressions, solving equations, and proving other mathematical concepts. In our integral, we used the identity \( \cos x = 2\cos^2 \frac{x}{2} - 1 \) to transform the expression into something integrable.

• Common examples are the Pythagorean identities, such as \( \sin^2 x + \cos^2 x = 1 \).
• These identities can convert complicating factors into more workable forms, especially for integrals.

By employing trigonometric identities, you reduce the complexity of the integral, which, in turn, makes the computation much more manageable. This step was crucial in our solution, as it allowed us to simplify \( \sqrt{1+\cos x} \) to a function that we could directly integrate.

Definite Integrals

Definite Integrals are a fundamental concept in calculus used to find the exact accumulation of quantities, like areas under a curve. A definite integral comes with limits of integration, indicating the range of the integral, as was the case with our exercise ranging from \( 0 \) to \( \pi \).

• The basic property involves integrating a function within a closed interval to get a numerical result.
• Definite integrals are crucial in physics, engineering, and probability for precise calculations.
• The integral of a function \( f(x) \) from \( a \) to \( b \) is denoted as \( \int_a^b f(x)\, dx \).

The process involves finding the antiderivative of the function and then evaluating it at the upper and lower limits. In this exercise, after simplifying the trigonometric integral, applying the limits of integration helped us determine that the accumulated area, or result of the definite integral, is \( 4 \). This mathematical tool is vital for exact calculations, ensuring accuracy and reliability in results.