Question

Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. $$\int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{2} x}$$

Short answer

Question: Evaluate the definite integral $\int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{2} x}$ using both a symbolic method and a numerical method, and compare the results. Answer: The symbolic method gives the exact value of the definite integral, $\frac{\pi}{4}$. The numerical method, using Simpson's rule, provides an approximation close to this value, which can be more or less accurate depending on the number of intermediate points used. Overall, the symbolic method provides an exact answer, while the numerical method offers an approximation that can be improved with finer techniques.

Step-by-step solution

Symbolic Method to evaluate integral

To evaluate the integral symbolically, first we will find an antiderivative for the function $$\frac{1}{1+\tan ^{2} x}$$. Recall the Pythagorean identity: $$\sec^2 x = 1 + \tan^2 x$$ Then, we can rewrite our integrand as: $$\frac{1}{\sec^2 x} = \cos^2 x$$ Now we can find an antiderivative: $$\int_{0}^{\pi/2} \cos^2 x \, dx$$ Using the power-reduction formula, we can rewrite $$\cos^2 x$$ as: $$\frac{1}{2} + \frac{1}{2}\cos(2x)$$ Now, integrate: $$\int_{0}^{\pi/2} \left(\frac{1}{2} + \frac{1}{2}\cos(2x)\right) dx = \frac{1}{2}\int_{0}^{\pi/2} dx + \frac{1}{2}\int_{0}^{\pi/2} \cos(2x) dx$$ $$= \frac{1}{2}\left[x\right]_0^{\pi/2} + \frac{1}{2}\left[\frac{1}{2}\sin(2x)\right]_0^{\pi/2}$$ $$= \frac{1}{2}\left(\frac{\pi}{2}\right) + \frac{1}{2}\left(\left[\frac{1}{2}\sin(\pi)\right] - \left[\frac{1}{2}\sin(0)\right]\right)$$ $$= \frac{\pi}{4}$$

Numerical Method to evaluate integral

Now, let's use a numerical method to approximate the definite integral value. We will use the Simpson's rule. Simpson's rule is a technique that fits a quadratic function to the data points and estimates the definite integral based on the average of several intermediate points. Using a Computer Algebra System, such as Mathematica or Python with the SciPy library, you can now apply Simpsons's rule to approximate the integral value. For example, in Python: ```python import numpy as np from scipy.integrate import simpson def integrand(x): return np.cos(x)**2 x_points = np.linspace(0, np.pi/2, 1000) y_points = integrand(x_points) approx_integral = simpson(y_points, x_points) ``` After running this code, you'll get the numerical approximation of the integral.

Compare the results

Now, it's time to compare the results from the symbolic and numerical methods. The symbolic method gave us the exact value of the definite integral: $$\int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{2} x} = \frac{\pi}{4}$$ And the numerical method, using Simpson's rule, should give an approximation close to this exact value. Depending on the number of intermediate points used in the numerical approximation, the numerical result may be more or less accurate. In conclusion, the symbolic method provides an exact value for the definite integral, while the numerical method gives an approximation that can be improved by using more intermediate points or using more advanced approximation techniques.

Key concepts

Symbolic Integration

Symbolic integration is a technique used to find the exact value of an integral by transforming the integrand into a simpler form for which an antiderivative can be easily determined. This contrasts with numerical methods, which provide approximate solutions. In our exercise, we used symbolic integration to find the exact value of the definite integral \( \int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{2} x} \).
This process involved recognizing the relationship between trigonometric identities. Specifically, we used the Pythagorean identity \( \sec^2 x = 1 + \tan^2 x \) to transform the integrand into a more manageable form \( \cos^2 x \).
By further using the power-reduction formula, we were able to express \( \cos^2 x \) as \( \frac{1}{2} + \frac{1}{2}\cos(2x) \) which is straightforward to integrate. The antiderivative provided an exact value of \( \frac{\pi}{4} \) for the integral.

Numerical Methods

Numerical methods are techniques used to approximate the solutions to mathematical problems that may be difficult to solve analytically. In calculus, numerical methods are vital for approximating definite integrals when an exact symbolic solution is complex or impossible to find.
These methods involve algorithms that compute the integral's approximate value based on function evaluations at specific points.

• Some common numerical integration methods include the Trapezoidal Rule, Simpson's Rule, and the Midpoint Rule.
• They vary in complexity and accuracy, often needing more computational resources to improve precision.

In our example, we applied a numerical method—Simpson's Rule—to approximate the integral \( \int_{0}^{\pi / 2} \cos^2 x \, dx \). By using a computer algebra system to evaluate the integral over many data points, the numerical approximation was closely aligned with the symbolic solution, \( \frac{\pi}{4} \). The more points we evaluate, the more accurate our approximation becomes.

Pythagorean Identities

Pythagorean identities are fundamental trigonometric relationships that express functions of angles in terms of the square of sine and cosine functions. These identities are vital for transforming complex trigonometric expressions into simpler, more integrable forms.
In our exercise, the Pythagorean identity \( \sec^2 x = 1 + \tan^2 x \) was key to simplifying the integrand of our definite integral problem. By recognizing this identity, we could convert \( \frac{1}{1+\tan ^{2} x} \) to \( \cos^2 x \), facilitating the process of finding an antiderivative.

• Pythagorean identities are: \( \sin^2 x + \cos^2 x = 1 \), \( 1 + \tan^2 x = \sec^2 x \), and \( 1 + \cot^2 x = \csc^2 x \).
• These identities help simplify and solve various trigonometric equations.

Understanding and applying these identities is crucial in symbolic integration, as demonstrated in our solution.

Simpson's Rule

Simpson's Rule is a numerical method used to approximate the value of definite integrals. This technique is particularly effective for functions that are well-behaved, such as those that are smooth and continuous over the interval of integration.
Simpson's Rule estimates the area under a curve by fitting parabolas to segments of the curve and calculating these areas. It provides a more accurate approximation compared to simpler methods like the Trapezoidal Rule by considering the curvature of the function rather than just linear approximations.
In our exercise, we employed Simpson's Rule to approximate the integral \( \int_{0}^{\pi / 2} \cos^2 x \, dx \). With a computer algebra system, we implemented this technique by evaluating the integrand at numerous equally spaced points, refining our approximation. The result was an integral value nearly equivalent to that found using symbolic integration.

• The accuracy of Simpson's Rule increases with the number of subdivisions, resulting in a more precise estimation.
• Optimal for scenarios where analytical solutions are challenging to derive.

Using Simpson's Rule can provide valuable insights into solving integrals numerically.