Use a power series to approximate the definite integral, I, to six decimal places: the integral from 0 to 0.3 of ((x^5)/(1+x^4))dx

Question

NEED HELP ASAP! please provide step by step solution, especially on how to estimate the integral to 6 decimal places. How can you do so without a calculator?

Answer 1

I am not sure what is wanted here, but I will suggest something.

To get the power series, divide the rational fraction by long division to get

x⁵(1 - x⁴ + x⁸ - x¹² + ...).

This power series converges for |x|<1 and the error committed does not exceed the first omitted term and you can integrate term by term. The power terms are not difficult without a calculator since you can just multiply sequentially by 3 and watch your decimal point carefully.!

Please note: the usual way of integrating this function would be to convert it to

x - [x/(1+x⁴)] and integrate the fraction as an arctan.