{Use of Tech} Approximating definite integrals 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.
∫₀⁰ᐧ³⁵ tan ⁻¹x dx

Verified step by step guidance

1. Identify the function to integrate: here it is \( f(x) = \tan^{-1}(x) \), also known as the inverse tangent or arctangent function.
2. Recall the Taylor series expansion of \( \tan^{-1}(x) \) centered at 0, which is \( \tan^{-1}(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \).
3. Set up the integral of the Taylor series term-by-term from 0 to 0.35: \( \int_0^{0.35} \tan^{-1}(x) \, dx = \int_0^{0.35} \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} \, dx = \sum_{n=0}^{\infty} (-1)^n \frac{1}{2n+1} \int_0^{0.35} x^{2n+1} \, dx \).
4. Integrate each term: \( \int_0^{0.35} x^{2n+1} \, dx = \frac{(0.35)^{2n+2}}{2n+2} \). Substitute back to get the series approximation for the integral: \( \sum_{n=0}^{\infty} (-1)^n \frac{(0.35)^{2n+2}}{(2n+1)(2n+2)} \).
5. Determine how many terms to keep by estimating the remainder (error) term of the alternating series to ensure the error is less than \( 10^{-4} \). Stop adding terms once the absolute value of the next term is smaller than \( 10^{-4} \).

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It allows approximation of functions like arctan(x) by polynomials, which are easier to integrate. The accuracy improves as more terms are included.

Definite Integral Approximation Using Series

Integrating a function approximated by its Taylor series term-by-term converts the integral into a sum of integrals of polynomials. This method simplifies evaluating definite integrals, especially when the exact integral is difficult to find analytically.

Error Estimation and Convergence Criteria

When approximating with Taylor series, it is crucial to estimate the remainder (error) to ensure the approximation meets a desired accuracy, here less than 10⁻⁴. Understanding convergence and bounding the error term guarantees the approximation's reliability.