Question

A remarkable integral \(1 \mathrm{t}\) is a fact that \(\int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{m} x}=\frac{\pi}{4},\) for all real numbers \(m\). a. Graph the integrand for \(m=-2,-3 / 2,-1,-1 / 2,0,1 / 2,1\) 3/2, and 2, and explain geometrically how the area under the curve on the interval \([0, \pi / 2]\) remains constant as \(m\) varies. b. Use a computer algebra system to confirm that the integral is constant for all \(m\).

Short answer

** Answer: The integral \(\int_{0}^{\pi / 2}\frac{dx}{1+\tan^m x}\) equals \(\frac{\pi}{4}\) for all real numbers \(m\) because, geometrically, the area under the curve remains constant in the interval \([0, \pi/2]\) regardless of the value of \(m\). This constant area represents the sum of all the values of the function multiplied by a small width in the given interval. Using a computer algebra system confirms this result for different values of \(m\).

Step-by-step solution

Draw graphs for different values of m

Draw the graph of the function \(\frac{1}{1 + \tan^m x}\) for \(m = -2, -3/2, -1, -1/2, 0, 1/2, 1, 3/2,\) and \(2\). Make sure to plot the graphs in the interval \([0, \pi / 2]\). This step is important to visualize the behavior and geometry of the integrands.

Analyze the geometrical properties of the graphs

Analyze the area under the curve for these graphs in the given interval \([0, \pi / 2]\). Observe that, regardless of the value of m, the area remains constant. This means the sum of all the values of the function (heights of small rectangles under the curve) multiplied by a small width (width of the rectangles) remains constant in the given interval. You should recognize that this is the geometrical evidence that \(\int_{0}^{\pi / 2} \frac{dx}{1+\tan^m x} = \frac{\pi}{4}\) for all real numbers \(m\).

Use a computer algebra system to confirm the result

Using a computer algebra system like Wolfram Alpha, Desmos, or Python's sympy library, calculate the integral for different values of m, especially the values provided in the question. Confirm that the integral indeed evaluates to \(\frac{\pi}{4}\), regardless of the value of \(m\). This important step allows you to verify the result and strengthen your understanding of the geometrical properties observed in the graphs.

Key concepts

Integral Properties

Understanding the properties of definite integrals is essential for calculus students. One fundamental property is that the value of a definite integral remains constant over a particular interval if the integrand—the function being integrated—is symmetric with respect to that interval.

In the given exercise, the remarkable fact that \[\int_{0}^{\pi / 2} \frac{d x}{1+\tan^{m} x}=\frac{\pi}{4},\] for all real numbers \(m\) showcases this property. Since the integration is done from \(0\) to \(\frac{\pi}{2}\), we are working within the first quadrant, where the trigonometric function tangent, and consequently \(\tan^{m} x\), exhibits certain symmetries.

Furthermore, a property of definite integrals is that the integral of an even function (a function that is symmetric about the y-axis), over an interval symmetric about the origin, is twice the integral from 0 to the half-length of the interval. Although \(\tan^m x\) is not always an even function, the manipulation through the power \(m\) and reciprocation in the integrand somehow preserves the area under the curve within the given interval. These intriguing properties of integral calculus highlight the surprising constancy and patterns that emerge within the discipline.

Computer Algebra Systems

Computer algebra systems (CAS) are powerful tools that extend beyond simple numerical calculations, helping students and mathematicians symbolically manipulate mathematical expressions and solve complex problems. In regards to our exercise, a CAS can be used to confirm that the integral \[\int_{0}^{\pi / 2} \frac{d x}{1+\tan^{m} x}\] remains \(\frac{\pi}{4}\) for all real numbers \(m\).

By inputting the integral into a CAS like Wolfram Alpha or using Python's sympy library, the student can verify the result for various values of \(m\) without relying solely on geometric intuition or manual computation. CAS not only provide a confirmation of the mathematical properties but also give students a hands-on approach to exploring and understanding calculus concepts. Such systems illustrate how technology can enhance learning and comprehension in mathematics, allowing for immediate exploration of difficult problems and visualization of concepts.

Geometric Interpretation of Integrals

The geometric interpretation of integrals is an integral part of understanding calculus. It helps students visualize the concepts and connect them to real-life applications. Definite integrals, in particular, represent the area under the curve of a function within a specific interval on the x-axis.

In the context of the given exercise problem, the integral \[\int_{0}^{\pi / 2} \frac{d x}{1+\tan^{m} x}=\frac{\pi}{4}\] for various powers of \(m\) reveals an interesting geometric property: the area under the curve, in this case, remains constant despite the changing shape of the graph as \(m\) varies. This demonstrates a unique case where the geometry of the function's graph has no impact on the integral's value between \(0\) and \(\pi / 2\).

Through graphing the integrand for different values of \(m\), students can observe firsthand that the area encapsulated by the curve maintains equality across these variations. This concept helps demystify the abstract nature of integrals by connecting them to tangible, visual phenomena, thus facilitating a more profound understanding of calculus.