Question

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}\). $$\int_{0}^{0.4} \ln \left(1+x^{2}\right) d x$$

Short answer

Answer: The approximate value of the integral is 0.020531209.

Step-by-step solution

Find the Taylor series representation of \(\ln(1+x^2)\)

To find the Taylor series representation, we need to expand \(\ln(1+x^2)\) with respect to \(x\). We will use the Taylor series expansion for natural logarithm: $$\ln(1+x) = x - \frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+\cdots$$ Plugging in \(x^2\) in place of \(x\), we get: $$\ln(1+x^2) = x^2 - \frac{1}{2}x^4+\frac{1}{3}x^6-\frac{1}{4}x^8+\cdots$$

Determine the number of terms for an error less than \(10^{-4}\)

To ensure the error remains within the desired bound, we need to find the first few terms of the Taylor series that add up to an error of less than \(10^{-4}\). For simplicity, we will analyze the worst-case scenario, i.e., the highest term contributing to the error by choosing \(x\) at the upper limit of integration, \(x = 0.4\). We will look for the first term such that the absolute value of that term will be less than \(10^{-4}\): $$\left|\frac{(-1)^n}{2n+1}(0.4)^{2n+2}\right| < 10^{-4}$$ Since the terms in the Taylor series alternate in sign, the absolute value of each term is decreasing. Thus it is enough to find the first \(n\) such that: $$\frac{(0.4)^{2n+2}}{2n+1} < 10^{-4}$$ By trying different values of \(n\), we discover that \(n = 5\) satisfies the inequality: $$\frac{(0.4)^{12}}{11} \approx 9.43 \times 10^{-5} < 10^{-4}$$ So we need to consider the terms up to \(n = 5\) in the Taylor series expansion to ensure the error is less than \(10^{-4}\): $$\ln(1+x^2) \approx x^2 - \frac{1}{2}x^4+\frac{1}{3}x^6-\frac{1}{4}x^8+\frac{1}{5}x^{10}$$

Integrate the Taylor series over the given limits

Now, we will integrate the Taylor series expansion derived above over the given limits \(0\) and \(0.4\). The definite integral will be: $$\int_{0}^{0.4} \ln \left(1+x^{2}\right) d x \approx \int_{0}^{0.4} \left( x^2 - \frac{1}{2}x^4+\frac{1}{3}x^6-\frac{1}{4}x^8+\frac{1}{5}x^{10} \right) dx$$ Integrating term by term: $$\int_{0}^{0.4} \left( x^2 - \frac{1}{2}x^4+\frac{1}{3}x^6-\frac{1}{4}x^8+\frac{1}{5}x^{10} \right) dx = \left[ \frac{1}{3}x^3 - \frac{1}{10}x^5 + \frac{1}{21}x^7 - \frac{1}{36}x^9 + \frac{1}{55}x^{11} \right]_{0}^{0.4}$$ Evaluating the result at the limits, we get: $$\approx \frac{1}{3}(0.4)^3 - \frac{1}{10}(0.4)^5 + \frac{1}{21}(0.4)^7 - \frac{1}{36}(0.4)^9 + \frac{1}{55}(0.4)^{11}$$

Compute the approximate value of the integral

Now, we will plug in the values and compute the approximate value of the integral: $$\approx 0.021333333 - 0.0008192 + 0.000017463 - 6.52 \times 10^{-7} + 4.07 \times 10^{-9}$$ $$\approx 0.020531209$$ Thus, the approximate value of the integral, with error less than \(10^{-4}\), is: $$\int_{0}^{0.4} \ln \left(1+x^{2}\right) d x \approx 0.020531209$$

Key concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus that lets us find the area under a curve within a specific interval. In other words, it's the total sum accumulated over that range. If you have a curve described by a mathematical function, a definite integral finds the total change between two points on this curve.
In our example, this means finding the total area between the function \ln(1+x^2)\ and the x-axis, from x = 0 to x = 0.4. We represent this as:

• \( \int_{0}^{0.4} \ln \left(1+x^{2}\right) \, dx \)

Unlike indefinite integrals, which provide a general formula, definite integrals give a specific numerical result. This makes them especially useful when dealing with computations and real-world applications.

Error Estimation

Error estimation is crucial when approximating integrals, especially when using methods like Taylor series because you're often using polynomial expressions to estimate more complex functions. It helps determine how accurate our approximation is.
When approximating our integral with a Taylor series, we need a way to ensure that the error, the difference between the actual value and our approximation, stays within a desired limit. In our task, the goal was to maintain the error below \(10^{-4}\).
To find the error, we looked at the highest degree term that contributes significantly to the error for given conditions—basically a way to understand the impact of each added term:

• The inequality used was:
  \( \left|\frac{(0.4)^{2n+2}}{2n+1}\right| < 10^{-4} \)

This inequality helps determine how many terms of the series are needed to get an accurate approximation. The smaller the resulting value, the less error in the approximation.

Natural Logarithms

The natural logarithm, often written as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It shows up frequently in calculus and mathematical modeling because of its natural relationship with exponential growth processes.
In the given exercise, we focus on \( \ln(1+x^2) \) which serves as our function to integrate. To make this task smaller and manageable, we use the Taylor series expansion. This allows us to represent the natural logarithm as an infinite sum of powers of \( x^2 \):

• \( \ln(1+x^2) = x^2 - \frac{1}{2}x^4 + \frac{1}{3}x^6 - \frac{1}{4}x^8 + \dots \)

This expansion is key because it lets us integrate term by term, making the process of solving the definite integral significantly simpler.

Polynomial Approximation

Polynomial approximation is a method of estimating more complex functions with simpler polynomial functions. In calculus, especially when dealing with integrals, these approximations can simplify complicated calculations.
Using the Taylor series, in our exercise, we transformed \( \ln(1+x^2) \) into a polynomial. This conversion allows us to integrate piece by piece,
significantly reducing the complexity. For example, breaking the natural logarithm into a series:

• \( x^2 - \frac{1}{2}x^4 + \frac{1}{3}x^6 - \frac{1}{4}x^8 + \frac{1}{5}x^{10}\)

These polynomial forms, while approximations, are very effective for calculating values over small intervals. The power of polynomial approximation is especially poignant with tools like the Taylor series, enabling us to not only approximate but also control and understand the error in our estimations.