Question

Use numerical methods or a calculator to approximate the following integrals as closely as possible. $$\int_{0}^{\pi / 2} \ln (\sin x) d x=\int_{0}^{\pi / 2} \ln (\cos x) d x=-\frac{\pi \ln 2}{2}$$

Short answer

Short answer: Using numerical integration methods such as the Trapezoidal rule, we can approximate the given integrals: $$\int_{0}^{\pi / 2} \ln (\sin x) d x$$ and $$\int_{0}^{\pi / 2} \ln (\cos x) d x$$ by dividing the integration interval into \(n\) equal subintervals and summing the areas of the trapezoids formed. By increasing the value of \(n\), we can obtain an approximation closer to the given value \(-\frac{\pi \ln 2}{2}\), confirming the validity of our results.

Step-by-step solution

Understand the integrals

Both integrals are definite integrals over the interval \([0, \pi/2]\). They involve natural logarithmic functions of trigonometric functions, which makes them more challenging to solve analytically. However, we can use numerical methods to estimate their values.

Choose a numerical integration method

There are various numerical integration methods that can be used to approximate the value of an integral, like the Trapezoidal rule, Simpson's rule, or Romberg integration. In this exercise, we will use the Trapezoidal rule, a simple and easy-to-use method. However, the method can be easily replaced with any other numerical integration method if needed.

Apply the Trapezoidal rule to the first integral

Using the Trapezoidal rule, we can approximate the value of the integral by dividing the interval into \(n\) equally-sized subintervals, and then summing the areas of the trapezoids that are formed. For the first integral, $$\int_{0}^{\pi / 2} \ln (\sin x) d x$$ we will divide the interval \([0, \pi/2]\) into \(n\) equally-sized subintervals and use the Trapezoidal rule to estimate the integral: $$T_n = \frac{h}{2}\Big[ \ln (\sin x_0) + 2 \ln (\sin x_1) + \cdots + 2 \ln (\sin x_{n-1}) + \ln (\sin x_n) \Big]$$ where \(h = \frac{\pi/2 - 0}{n}\), and \(x_i = 0 + ih\) for \(i = 0, 1, 2, \dots, n\). Using a calculator or a software tool, we can compute \(T_n\) for different values of \(n\), until we reach an approximation that is close to the given value of the integral, \(-\frac{\pi \ln 2}{2}\).

Apply the Trapezoidal rule to the second integral

Similarly, for the second integral, $$\int_{0}^{\pi / 2} \ln (\cos x) d x$$ we will divide the interval \([0, \pi/2]\) into \(n\) equally-sized subintervals and use the Trapezoidal rule to estimate the integral: $$T_n = \frac{h}{2}\Big[ \ln (\cos x_0) + 2 \ln (\cos x_1) + \cdots + 2 \ln (\cos x_{n-1}) + \ln (\cos x_n) \Big]$$ where \(h = \frac{\pi/2 - 0}{n}\), and \(x_i = 0 + ih\) for \(i = 0, 1, 2, \dots, n\). As before, we will use a calculator or a software tool to compute \(T_n\) for different values of \(n\), until we reach an approximation that is close to the given value of the integral, \(-\frac{\pi \ln 2}{2}\).

Conclusion

By applying the Trapezoidal rule to both integrals and comparing our approximations to the given value of \(-\frac{\pi \ln 2}{2}\), we can verify the approximated value of these integrals using numerical methods. This process can be repeated with other numerical integration methods to see if they yield even more accurate approximations.

Key concepts

Trapezoidal Rule

The Trapezoidal Rule is a straightforward numerical method to approximate the definite integral of a function. It's particularly useful when dealing with complex integrals that are difficult to solve analytically. The method works by dividing the integration interval into smaller subintervals. Each subinterval is treated like a trapezoid. The area of all these trapezoids is then summed up to approximate the total area under the curve.

To apply the Trapezoidal Rule, consider the function you are integrating over the interval \([a, b]\). You divide this interval into \(n\) equal parts, each with a width of \((h = \frac{b-a}{n})\). Calculate the function's value at each of these points. Then, use the formula:

• \(T_n = \frac{h}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]\)

This method estimates areas by creating trapezoids between each pair of consecutive points, thus the name. It's an effective approach especially in scenarios where a precise analytical solution is complex.

Definite Integrals

Definite integrals are a core concept in calculus used to calculate the area under a curve between two points on the x-axis. Specifically, a definite integral of a function \(f(x)\) over an interval \([a, b]\) is denoted as \(\int_{a}^{b} f(x) \, dx\).

This integral has a boundary defined by the limits \(a\) and \(b\). It gives you an exact value representing the accumulated quantity—such as area or volume. In the given exercise, the task was to approximate definite integrals involving natural logarithms of trigonometric functions in the interval \([0, \pi/2]\).

Computing definite integrals analytically could be challenging due to the algebraic complexity of the functions involved. This is where numerical methods like the Trapezoidal Rule come in handy, allowing for approximation and thus enabling solutions to problems that may otherwise be difficult to solve analytically.

Natural Logarithms

Natural logarithms are logarithms to the base \(e\), where \(e\) is a mathematical constant approximately equal to 2.71828. The natural logarithm of a number \(x\) is usually denoted as \(\ln(x)\).

Natural logarithms have unique properties that make them very useful in mathematics, especially calculus. For instance, they simplify the differentiation of exponential functions and provide insights into problems involving decay, growth, and other processes involving ratios.

When integrating functions like \(\ln(\sin x)\) or \(\ln(\cos x)\) over certain intervals, numerical methods are often used to approximate solutions. These topics often appear in calculus problems where the solution requires a blend of understanding both natural logs and the integral calculus concepts.

Trigonometric Functions

Trigonometric functions include basic functions like sine, cosine, and tangent used widely in mathematics to model periodic phenomena.

Trigonometric functions are important because they describe relationships in triangles, particularly right-angled ones, and are a foundation for simple harmonic motion and wave behavior.

In the problem at hand, integrals involve natural logarithms of sine and cosine functions. These integrals surprisingly converge to the same value despite the functions being different, illustrating the fascinating nature of calculus and the symmetry in these specific trigonometric relationships.

Calculating integrals involving these functions numerically, as seen with the Trapezoidal Rule application in this exercise, ensures students and professionals alike can manage complex computations efficiently even when analytic solutions are elusive.