I am trying to simplify an integration expression
$$\int_0^{D}\frac{1}{x^{a}+1}dx$$
I know that for $a=2$, I can have it as $\arctan(x)$. But what about the general case?
Thanks for your help!
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2What about PFD , Partial Fraction Decomposition ? – Cardinal Dec 04 '15 at 14:12
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How can I decompose it? – KennyYang Dec 04 '15 at 14:14
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Just Read a Mathematics Textbook, By the way you can try this http://www.wolframalpha.com/widgets/view.jsp?id=ec4a062bb304f88c2ba0b631d7acabbc – Cardinal Dec 04 '15 at 14:17
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Is $a$ an integer? – Bernard Dec 04 '15 at 14:17
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not sure, $a$ might not be an integer. – KennyYang Dec 04 '15 at 14:21
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I think wolfram cannot solve my problem with general $a$. – KennyYang Dec 04 '15 at 14:24
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2If you want the most general answer for any real $a$, it is given in terms of Hyper Geometric function $I = \int {{1 \over {{x^a} + 1}} = } x,,{}_2^{}F_1^{}\left( {1,{1 \over a};1 + {1 \over a}; - {x^a}} \right)$ – Hosein Rahnama Dec 04 '15 at 16:08
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1For positive integer values of $a,$ see Solving this integral?. – Dave L. Renfro Dec 04 '15 at 16:30
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1@Cardinal Your suggestion of partial fractions is not useful. Prove me wrong with a full answer for general $a$ or even just arbitrary positive integer $a$ if you think otherwise. – Simon S Dec 04 '15 at 18:13
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@I didn't assert that it is the most general case. This integral can be solved through confluent hypergeometric function, as mentioned by H.R – Cardinal Dec 04 '15 at 19:21