Are there any integrals that can't be solved with only trig substitution? An integral that requires you to use a hyperbolic or inverse hyperbolic substitution?
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@symplectomorphic By solve, I mean find the antiderivative. And can't your example by solved using a tan substitution? – dfg Apr 18 '14 at 20:43
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No there aren't, but hyperbolic substitutions are frequently preferable to (easier than) trigonometric substitutions. – ryang Jan 08 '22 at 19:05
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One way of viewing the hyperbolic functions is that they are "abbreviations" for combinations of exponential functions, so whatever can be done using hyperbolic functions can equally be done using exponentials - albeit in a less concise manner. Similarly, inverse hyperbolics are really just log functions in various combinations.
What this means is that if an integral can be done with a hyperbolic substitution, then it can also be done without hyperbolics, by means of a (possibly more complicated) substitution involving exponential functions.
Old John
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Thank you. Are their any integrals that can be solved using a hyperbolic/inverse hyperbolic substitution but not with a trig substitution> – dfg Apr 18 '14 at 20:51
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Well, since hyperbolic functions are really the same as trigonometric functions but just rotated by multiplication by $i$ in the complex plane, I would say no. – Old John Apr 18 '14 at 20:53
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