Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Somewhat inspired by this question about antiderivatives, I started to check whether or not that function had an elementary antiderivative. Then, after checking with Maxima, it struck me that, by simplifying the $\sec(x)$ and $\tan(x)$ terms using Taylor series, I could effectively solve the antiderivative of $\int {\sec\left(x\right)\tan\left(x\right) \over 3x + 5}\,{\rm d}x$.

Solving this, at origin 0 and with depth 8, I get the following expression.

$$ -\frac{9646207\,{x}^{9}}{1181250000}+\frac{9646207\,{x}^{8}}{630000000}-\frac{14069\,{x}^{7}}{875000}+\frac{14069\,{x}^{6}}{450000}-\frac{179\,{x}^{5}}{6250}+\frac{179\,{x}^{4}}{3000}-\frac{{x}^{3}}{25}+\frac{{x}^{2}}{10} $$

However, and back when I had Calculus, I never remembered using a Taylor series in order to solve an antiderivative.

Besides the resulting antiderivative being an approximation that degrades the further away the function is from the Taylor origin (as a Taylor series has a sort of implied error), what other faults or errors might happen should one use this technique?

share|improve this question
    
In a typical second-semester calculus course in the USA, integration using Taylor series appears at the end; it's sort of a pinnacle of the course, tying in integrals and infinite series. –  Post No Bulls Dec 31 '13 at 6:42
    
Yeah, that was a fun one... I still haven't figured it out. Guess my next foray is going to be Taylor Series. –  Chris Feb 4 at 19:18

1 Answer 1

up vote 1 down vote accepted

There is nothing wrong with using Taylor's series for anti-derivative. In fact, the very first breakthrough in the computation of $\pi$ came when Gregory used your idea to get the anti-derivative for $\arctan$. Gauss did the same for $\arccos$. You are in good company!

share|improve this answer
    
I have to say I have yet to find anything about Gauss and $arccos$. This being said, a man by the name of Gregory Chudnovsky, together with his brother, developed a fast algorithm to calculate $\pi$. I'll keep this for a day or two, if no one gives another answer, I'll accept this. –  Doktoro Reichard Dec 28 '13 at 23:03
    
The original proof of Sophomore's dream also used the sam idea as you did. Please see en.wikipedia.org/wiki/Sophomore%27s_dream. Chodonvsky's method is a bit different. –  user44197 Dec 29 '13 at 4:48

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.