How do you integrate hyperbolic trig functions?

Answer 1

Using their definitions is the simplest method for integrating (or differentiating) the hyperbolic functions:

#sinh(x)=(e^x-e^(-x))/2# #cosh(x)=(e^x+e^(-x))/2# #tanh(x)=sinh(x)/cosh(x)=(e^x-e^(-x))/(e^x+e^(-x))# #coth(x)=cosh(x)/sinh(x)=(e^x+e^(-x))/(e^x-e^(-x))#

It should be fairly simple to demonstrate from here that

#int sinh(x)dx = cosh(x) + C# #int cosh(x)dx = sinh(x) + C# #int tanh(x)dx = ln(cosh((x)) + C# #int coth(x)dx = ln(sinh(x))+ C#

where C is the integration constant. I'll display the first two of these here:

#int sinh(x)dx = int (e^x-e^-x)/2 = int e^x/2-e^(-x)/2dx# #=e^x/2-(-e^-x)/2 + C# (where #C# is the constant of integration) #=e^x/2+e^(-x)/2 + C# #=cosh(x)+C#.
Similarly, #int cosh(x)dx = int e^x/2+e^(-x)/2dx# #=e^x/2+(-e^-x)/2 + C# #=e^x/2-e^-x/2 + C# #=sinh(x) + C#.
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Answer 2

To integrate hyperbolic trigonometric functions, you can use the following formulas:

  1. Integral of hyperbolic sine (sinh): ∫ sinh(x) dx = cosh(x) + C

  2. Integral of hyperbolic cosine (cosh): ∫ cosh(x) dx = sinh(x) + C

  3. Integral of hyperbolic tangent (tanh): ∫ tanh(x) dx = ln|cosh(x)| + C

  4. Integral of hyperbolic secant (sech): ∫ sech(x) dx = ln|tanh(x/2) + sec(x)| + C

  5. Integral of hyperbolic cosecant (csch): ∫ csch(x) dx = ln|tanh(x/2) - cot(x/2)| + C

  6. Integral of hyperbolic cotangent (coth): ∫ coth(x) dx = ln|sinh(x)| + C

Remember to add the constant of integration (C) when integrating.

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Answer 3

To integrate hyperbolic trigonometric functions, you can use the following integration formulas:

  1. Integral of sinh(x): ∫sinh(x) dx = cosh(x) + C

  2. Integral of cosh(x): ∫cosh(x) dx = sinh(x) + C

  3. Integral of tanh(x): ∫tanh(x) dx = ln|cosh(x)| + C

  4. Integral of coth(x): ∫coth(x) dx = ln|sinh(x)| + C

  5. Integral of sech(x): ∫sech(x) dx = ln|tanh(x/2) + sech(x)| + C

  6. Integral of csch(x): ∫csch(x) dx = -ln|coth(x/2) + csch(x)| + C

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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