Integration involving hyperbolic functions, hyperbolic identities and hyperbolic substitutions
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Integration involving hyperbolic functions - Standard forms for the integrals that integrate
Integrals that give inverse hyperbolic functions Certain expressions integrate to give inverse hyperbolic functions. [Equation goes here - download the original to see it.] Proof We have shown that [Equation goes here - download the original to see it.] Hence, since integration is the inverse operation of differentiation. [Equation goes here - download the original to see it.] We also have [Equation goes here - download the original to see it.] We have already seen that [Equation goes here - download the original to see it.] On substituting (x/a) for x, and applying the chain rule: [Equation goes here - download the original to see it.] The remaining principal hyperbolic function gives the following form [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Applications of these formulae may be direct instances of substitution into them, or may require more insight and algebraic manipulation. Example [Equation goes here - download the original to see it.] Which is easily obtained by substituting a = 3 in [Equation goes here - download the original to see it.] However, in this second example matters are not so immediately clear. Example [Equation goes here - download the original to see it.] The 'trick' is to complete the square on x2 + x + 1 [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]
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Contents of
Integration involving hyperbolic functions, hyperbolic identities and hyperbolic substitutions
1 Integration involving hyperbolic functions - Standard forms for the integrals that integrate
2 Hyperbolic substitutions for the evaluation of definite or indefinite integrals
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