Calculate the integral in terms of the inverse hyperbolic functions. ∫(d x)/(√(9 x^2-4))

Name: Problem 18, Chapter 7: Calculus for AP
Uploaded: 2019-06-27T19:37:36-08:00
Duration: 1 min 3 s
Channel: Bahar Tehranipoor
Description: Calculate the integral in terms of the inverse hyperbolic functions. ∫(d x)/(√(9 x^2-4))

step 1 Let's evaluate this integral using what we know about inverse hyperbolic functions. So what we c

Calculate the integral in terms of the inverse hyperbolic...

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Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions, such as arccosh and arcsinh, are the inverses of the hyperbolic cosine and sine functions. They often appear as results of integration when dealing with integrals involving quadratic expressions under a square root. Their properties allow the antiderivatives to be expressed in a more compact and meaningful form.

Hyperbolic Substitution

Hyperbolic substitution is a technique used in integration to simplify expressions containing square roots of quadratic forms. By substituting a variable with a hyperbolic function and using identities like cosh²(u) - sinh²(u) = 1, the integral can be transformed into a standard form that is easier to evaluate.

Integration Involving Square Roots

Integrals that contain square roots of quadratic expressions, such as those of the form ?dx/?(ax² - b), require specific methods for solution. Recognizing the structure of the integrand allows one to choose an appropriate substitution, such as hyperbolic or trigonometric substitution, thus simplifying the expression for straightforward integration.

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