Question

use the power series to approximate the definite integral to 6 decimal places

Answer 1

\[\int\limits_{0}^{0.4}\ln(1+x ^{4})dx\]

Answer 2

Hint:
Expand ln(1+u) into a Maclaurin series, then substitute \(x^4\) for u.
As for the accuracy, since the resulting alternating series jumps in powers of 4 convergence to \(10^{-6}\) is rapid within 4-5 terms.

Answer 3

As a matter of fact, 3 terms are sufficient for an accuracy of 6 significant figures, nice!