Use two terms of a Maclaurin series to estimate this integral, and predict your error using the Alternating Series Estimation Theorem.

Question

∫ x^2 * sin(x^2) dx

Use two terms of a Maclaurin series to estimate this integral, and predict your error using the Alternating Series Estimation Theorem.

Please include formulas so I can follow along better. Please and thank you.

Answer 1

sin(x0 = x - x³/3! + x⁵/5!..

sin(x²) = x² - x⁶/2 + x¹⁰/4!

x²sin(x²) = x⁴ - x⁸/2 + x¹²/4!

The two terms (do the third as well) of the Maclaurin Series of x²sin(x²) can be integrated

The error is less than the next term's coefficient, so you need the 3rd term to bound the error.