Maclaurin Series

Question

How do you derive the Maclaurin series for f(x) = integral (sinx / x) dx? And what are the first 4 nonzero terms?

Answer 1

You definitely don't want to use the Maclaurin formula for this: If F(x) is the integral and f(x) is sin(x)/x, you get

F(x) = F(0) + f(0)x + f'(0)x² + f''(0)x³ +...

The derivatives get crazy quickly and every other one is zero. You can work out the terms are 0 + x + 0 +...

You can use the sin(x) series and divide by x and integrate term-by-term:

sin(x) = x - x³/3! + x⁵/5! - x⁷/7!

sin(x)/x = 1 - x²/3! + x⁴/5! - x⁶/7!

Now, take the integral of each term and you will get the first four terms of the series.