Hyperbolic functions such as \\(\sinh{x}\\), \\(\cosh{x}\\), and \\(\tanh{x}\\) are crucial in mathematics, mirroring trigonometric functions but based on hyperbolas. Their integrals are essential in calculus, with applications ranging from modeling catenary curves to solving complex problems in physics and engineering. Advanced techniques like integration by parts and substitution, as well as the integration of inverse hyperbolic functions, play a significant role in addressing real-world issues in various scientific fields.
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Hyperbolic functions are mathematical relations defined in the context of a hyperbola, similar to trigonometric functions
Hyperbolic Sine
Hyperbolic sine is represented by \(\sinh{x}\) and is used in various mathematical applications, such as modeling the catenary curve
Hyperbolic Cosine
Hyperbolic cosine is represented by \(\cosh{x}\) and is used in various mathematical applications, such as modeling the catenary curve
Hyperbolic Tangent
Hyperbolic tangent is represented by \(\tanh{x}\) and is used in various mathematical applications, such as modeling the catenary curve
Hyperbolic Secant
Hyperbolic secant is represented by \(\sech{x}\) and is used in various mathematical applications, such as modeling the catenary curve
Hyperbolic Cosecant
Hyperbolic cosecant is represented by \(\csch{x}\) and is used in various mathematical applications, such as modeling the catenary curve
Hyperbolic Cotangent
Hyperbolic cotangent is represented by \(\coth{x}\) and is used in various mathematical applications, such as modeling the catenary curve
Integration is a fundamental operation in calculus used to determine the area under a curve
Hyperbolic Sine
The integral of hyperbolic sine is given by \(\int \sinh{x} \,dx = \cosh{x} + C\)
Hyperbolic Cosine
The integral of hyperbolic cosine is given by \(\int \cosh{x} \,dx = \sinh{x} + C\)
Hyperbolic Tangent
The integral of hyperbolic tangent is given by \(\int \tanh{x} \,dx = \ln{|\cosh{x}|} + C\)
Hyperbolic Secant
The integral of hyperbolic secant is given by \(\int \sech{x} \,dx = 2\tan^{-1}{(\tanh{\frac{x}{2}})} + C\)
Hyperbolic Cosecant
The integral of hyperbolic cosecant is given by \(\int \csch{x} \,dx = \ln{\left| \tanh{\frac{x}{2}} \right|} + C\)
Hyperbolic Cotangent
The integral of hyperbolic cotangent is given by \(\int \coth{x} \,dx = \ln{\left| \sinh{x} \right|} + C\)
Integration by Parts
Integration by parts is a useful method for integrating products of hyperbolic functions with other functions, such as polynomials
Substitution
Substitution is a powerful technique for simplifying integrals involving hyperbolic functions by replacing a segment with a known hyperbolic identity
Inverse Hyperbolic Sine
The integral of inverse hyperbolic sine is given by \(\int \sinh^{-1}{x} \,dx = x \sinh^{-1}{x} - \sqrt{x^2 + 1} + C\)
Inverse Hyperbolic Cosine
The integral of inverse hyperbolic cosine is given by \(\int \cosh^{-1}{x} \,dx = x \cosh^{-1}{x} - \sqrt{x^2 - 1} + C\)
Inverse Hyperbolic Tangent
The integral of inverse hyperbolic tangent is given by \(\int \tanh^{-1}{x} \,dx = x \tanh^{-1}{x} + \frac{1}{2}\ln{\left(1-x^2\right)} + C\)
Integration of hyperbolic functions is essential in understanding the shape of a suspended cable, represented by the catenary curve
Integration of hyperbolic functions has practical significance in fields such as physics and engineering, particularly in problems related to heat transfer, wave propagation, and potential theory
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Hyperbolic functions are essential in mathematics, for instance, in modeling the ______ curve, depicting a hanging chain's shape under gravity.
catenary
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To integrate the hyperbolic cotangent function, one would use the formula ______.
ln(|sinh(x)|) + C
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Integration by parts: usage with hyperbolic functions
Decompose integrand into parts for separate differentiation and integration; useful for products of hyperbolic and polynomial functions.
