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Chapter 7: Problem 58

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. $$\int_{1 / 2}^{1} \frac{\sin ^{-1} x}{x} d x$$

Short Answer

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Question: Evaluate the definite integral $\int_{1 / 2}^{1} \frac{\sin^{-1} x}{x} d x$ using both a symbolic method and a numerical method, and compare the results. Answer: Using the symbolic method, the exact value of the integral is stored in `symbolic_result`. By applying the numerical method, the approximate value of the integral is stored in `numerical_result` along with an estimation of the error. Comparing the symbolic and numerical results will reveal the differences between these two methods.

Step by step solution

01

Symbolic Method

For the symbolic evaluation of the integral, we can use a computer algebra system (CAS) like "Wolfram Alpha" or "SymPy" in Python. Here, we'll demonstrate how to do this using SymPy in Python. First, we need to import the necessary libraries and define the variables: ```python import sympy as sp x = sp.Symbol('x') ``` Now, we can define the function and compute the integral. ```python f = (sp.asin(x)) / x symbolic_result = sp.integrate(f, (x, 1/2, 1)) ``` The exact value of the integral (obtained by the symbolic method) will be stored in the variable `symbolic_result`.
02

Numerical Method

For the numerical evaluation of the integral, we can use some numerical integration techniques like the Trapezoidal Rule or Simpson's Rule. Here, we'll demonstrate how to use `quad` function from SciPy library in Python for this purpose. First, we need to import the necessary libraries: ```python import numpy as np from scipy.integrate import quad ``` Now, we can define the function and compute the integral numerically. ```python def g(x): return (np.arcsin(x)) / x numerical_result, error = quad(g, 1/2, 1) ``` The approximate value of the integral (obtained by the numerical method) will be stored in the variable `numerical_result`, while `error` contains an estimation of the error.
03

Compare the results

Now that we have obtained the results using both the symbolic and the numerical methods, we can compare them by printing them out in a readable format. ```python print(f"Symbolic result: {symbolic_result}") print(f"Numerical result: {numerical_result}") ``` This will show the exact value and the approximate value calculated by the two different methods. In most cases, the results will be quite similar. However, depending on the complexity of the integral or the chosen numerical method, there might be slight differences between the results.

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Most popular questions from this chapter

Use integration by parts to evaluate the following integrals. $$\int_{0}^{1} x \ln x d x$$

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{2 x^{3}+x^{2}-6 x+7}{x^{2}+x-6} d x$$

By reduction formula 4 in Section 3 $$\int \sec ^{3} u d u=\frac{1}{2}(\sec u \tan u+\ln |\sec u+\tan u|)+C$$ Graph the following functions and find the area under the curve on the given interval. $$f(x)=\left(9-x^{2}\right)^{-2},\left[0, \frac{3}{2}\right]$$

Let \(R\) be the region between the curves \(y=e^{-c x}\) and \(y=-e^{-c x}\) on the interval \([a, \infty),\) where \(a \geq 0\) and \(c \geq 0 .\) The center of mass of \(R\) is located at \((\bar{x}, 0)\) where \(\bar{x}=\frac{\int_{a}^{\infty} x e^{-c x} d x}{\int_{a}^{\infty} e^{-c x} d x} .\) (The profile of the Eiffel Tower is modeled by the two exponential curves.) a. For \(a=0\) and \(c=2,\) sketch the curves that define \(R\) and find the center of mass of \(R\). Indicate the location of the center of mass. b. With \(a=0\) and \(c=2,\) find equations of the lines tangent to the curves at the points corresponding to \(x=0.\) c. Show that the tangent lines intersect at the center of mass. d. Show that this same property holds for any \(a \geq 0\) and any \(c>0 ;\) that is, the tangent lines to the curves \(y=\pm e^{-c x}\) at \(x=a\) intersect at the center of mass of \(R\) (Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique \(332(2004): 571-584 .)\)

Challenge Show that with the change of variables \(u=\sqrt{\tan x}\) the integral \(\int \sqrt{\tan x} d x\) can be converted to an integral amenabl to partial fractions. Evaluate \(\int_{0}^{\pi / 4} \sqrt{\tan x} d x\)
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