Question

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

Accepted Answer

Recall the Taylor series expansion for $$ \cos x $$ centered at 0, which is given by:

$$ \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots $$ Substitute the Taylor series for $$ \cos x $$ into the integral $$ \int_0^1 x \cos x \, dx . $$This gives:

$$ \int_0^1 x \cos x \, dx = \int_0^1 x \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \right) dx = \int_0^1 \left( x - \frac{x^3}{2!} + \frac{x^5}{4!} - \cdots \right) dx $$ Integrate the series term-by-term over the interval from 0 to 1:

$$ \int_0^1 x \, dx - \int_0^1 \frac{x^3}{2!} \, dx + \int_0^1 \frac{x^5}{4!} \, dx - \cdots = \left[ \frac{x^2}{2} \right]_0^1 - \frac{1}{2!} \left[ \frac{x^4}{4} \right]_0^1 + \frac{1}{4!} \left[ \frac{x^6}{6} \right]_0^1 - \cdots $$ Evaluate each definite integral at the limits 0 and 1, simplifying the expressions to get a series in terms of constants:

$$ \frac{1}{2} - \frac{1}{2! \cdot 4} + \frac{1}{4! \cdot 6} - \cdots $$ 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.

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

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Key Concepts

Taylor Series Expansion

Term-by-Term Integration of Power Series

Error Estimation in Taylor Series Approximations