How do you integrate hyperbolic trig functions?
Using their definitions is the simplest method for integrating (or differentiating) the hyperbolic functions:
It should be fairly simple to demonstrate from here that
where C is the integration constant. I'll display the first two of these here:
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To integrate hyperbolic trigonometric functions, you can use the following formulas:
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Integral of hyperbolic sine (sinh): ∫ sinh(x) dx = cosh(x) + C
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Integral of hyperbolic cosine (cosh): ∫ cosh(x) dx = sinh(x) + C
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Integral of hyperbolic tangent (tanh): ∫ tanh(x) dx = ln|cosh(x)| + C
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Integral of hyperbolic secant (sech): ∫ sech(x) dx = ln|tanh(x/2) + sec(x)| + C
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Integral of hyperbolic cosecant (csch): ∫ csch(x) dx = ln|tanh(x/2) - cot(x/2)| + C
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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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To integrate hyperbolic trigonometric functions, you can use the following integration formulas:
-
Integral of sinh(x): ∫sinh(x) dx = cosh(x) + C
-
Integral of cosh(x): ∫cosh(x) dx = sinh(x) + C
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Integral of tanh(x): ∫tanh(x) dx = ln|cosh(x)| + C
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Integral of coth(x): ∫coth(x) dx = ln|sinh(x)| + C
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Integral of sech(x): ∫sech(x) dx = ln|tanh(x/2) + sech(x)| + C
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Integral of csch(x): ∫csch(x) dx = -ln|coth(x/2) + csch(x)| + C
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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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