For the following indefinite integral, find the full power series centered at $x = 0$ and then give the first 5 nonzero terms of the power series.
$f (x) = \int \frac{e^{5 x} - 1}{8 x} d x$
$f (x) = C + \sum_{n = 1}^{\infty}$
$f (x) = C + ◻ + ◻ + ◻ + ◻ + ◻ + \dots$

Answer

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Answer

3. Evaluate integral: $f (x) = C + \frac{5}{8} - \frac{25 x}{48} + \frac{125 x^{2}}{144} - \frac{625 x^{3}}{1728} + \dots$

Steps

Step 1 ：1. Find Maclaurin series of $e^{5 x}$ : $e^{5 x} = \sum_{n = 0}^{\infty} \frac{(5 x)^{n}}{n!}$

Step 2 ：2. Plug Maclaurin series into integral: $f (x) = \int \frac{\sum_{n = 0}^{\infty} \frac{(5 x)^{n}}{n!} - 1}{8 x} d x$

Step 3 ：3. Evaluate integral: $f (x) = C + \frac{5}{8} - \frac{25 x}{48} + \frac{125 x^{2}}{144} - \frac{625 x^{3}}{1728} + \dots$