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.
\[
f(x)=\int \frac{e^{5 x}-1}{8 x} d x
\]
\[
f(x)=C+\sum_{n=1}^{\infty}
\]
\[
f(x)=C+\square+\square+\square+\square+\square+\cdots
\]

Answer

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Answer

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

Steps

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

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

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