Definite integrals by power series Use a Taylor series to approximate the following definite integrals. Retain as many terms as necessary to ensure the error is less than 10⁻³.
∫₀1 x cos x dx

Verified step by step guidance

1. Recall the Taylor series expansion for \( \cos x \) centered at 0, which is given by: \[ \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]
2. Substitute the Taylor series for \( \cos x \) into the integral \( \int_0^1 x \cos x \, dx \). This gives: \[ \int_0^1 x \cos x \, dx = \int_0^1 x \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \right) dx = \int_0^1 \left( x - \frac{x^3}{2!} + \frac{x^5}{4!} - \cdots \right) dx \]
3. Integrate the series term-by-term over the interval from 0 to 1: \[ \int_0^1 x \, dx - \int_0^1 \frac{x^3}{2!} \, dx + \int_0^1 \frac{x^5}{4!} \, dx - \cdots = \left[ \frac{x^2}{2} \right]_0^1 - \frac{1}{2!} \left[ \frac{x^4}{4} \right]_0^1 + \frac{1}{4!} \left[ \frac{x^6}{6} \right]_0^1 - \cdots \]
4. Evaluate each definite integral at the limits 0 and 1, simplifying the expressions to get a series in terms of constants: \[ \frac{1}{2} - \frac{1}{2! \cdot 4} + \frac{1}{4! \cdot 6} - \cdots \]
5. Determine how many terms to keep by estimating the error of the next term in the series. Continue adding terms until the absolute value of the next term is less than \( 10^{-3} \), ensuring the approximation meets the required accuracy.

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. For functions like cosine, the series allows approximation by polynomials, which simplifies integration and error estimation.

Term-by-Term Integration of Power Series

When a function is expressed as a power series, its definite integral over an interval can be found by integrating each term individually. This method converts complex integrals into sums of simpler polynomial integrals.

Error Estimation in Taylor Series Approximations

To ensure the approximation is accurate within a specified tolerance, the remainder or error term of the Taylor series must be estimated. This involves bounding the next omitted term to confirm the error is less than the desired threshold, such as 10⁻³.