Recipe
Integrate a non-elementary function using its power series
Use when: a function has no elementary antiderivative, but a power series for it is already known
Goal: either a series antiderivative, or a numeric approximation of a definite integral, accurate to a stated number of decimal places
Taught in Integrating non-elementary functions
Quick recipe
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1
Write down the already-derived Maclaurin (or other) series for the integrand.
Example
e^{-x^4} = \displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{4n}}{n!} = 1 - x^4 + \dfrac{x^8}{2} - \dfrac{x^{12}}{6} + \cdots
Why: term-by-term integration operates on the series, not on the original function directly, so the series has to be in hand first.
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2
Apply the ordinary power rule to each individual term of the series, adding one overall constant of integration at the end, not one per term.
Example
\displaystyle\int e^{-x^4}\,dx = \sum_{n=0}^{\infty}\frac{(-1)^n x^{4n+1}}{n!(4n+1)} + C = x - \frac{x^5}{5} + \frac{x^9}{18} - \frac{x^{13}}{78} + \cdots + C
Why: each term of the series is a plain power of x, so the same power rule that already works for polynomials applies to each one individually.
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3
Apply the Fundamental Theorem to the series antiderivative from Step 2, evaluated between the given limits, so the constant of integration cancels.
Example
\displaystyle\int_0^{0.5} e^{-x^4}\,dx = \left[x-\frac{x^5}{5}+\frac{x^9}{18}-\cdots\right]_0^{0.5}.
- only an indefinite antiderivative is requested — stop after Step 2.
- a definite integral, or a specific decimal accuracy, is requested — continue to Step 4.
Why: the series antiderivative from Step 2 is a genuine antiderivative, so the Fundamental Theorem applies to it exactly the way it applies to any other antiderivative.
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4
Keep adding terms until the next unused term is smaller than the required accuracy; for an alternating, decreasing series, that next term bounds the total error.
Example
At x=0.5: 0.5 - \dfrac{0.03125}{5} + \dfrac{0.001953}{18} - \cdots \approx 0.5 - 0.00625 + 0.0001085 \approx 0.4939, with the next unused term already under 0.00001, safely inside 4-decimal accuracy.
Why: the alternating series error bound gives a guaranteed accuracy without needing to sum infinitely many terms by hand.
Common mistakes
- Adding a separate constant of integration to every term instead of a single overall constant at the end.
- Applying the power rule's exponent-plus-one to the index n instead of to the actual exponent already present in each series term.
- Stopping the sum before the next unused term is actually smaller than the required accuracy, understating the true error.
- Using the alternating-series error bound on a series that isn't actually alternating, or whose terms aren't decreasing in size.