Question

Evaluate the integrals in Exercises 67–74 in terms ofa. inverse hyperbolic functions.71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))

Accepted Answer

Identify the integral to be evaluated: $$\int_{\frac{1}{5}}^{\frac{3}{13}} \frac{dx}{x \sqrt{1 - 16x^2$}}. $$Recognize that the integrand has the form $$\frac{1}{x \sqrt{1 - a^2$ x^2$}}$$ with $$a = 4$$, which suggests using a substitution or a known formula involving inverse hyperbolic functions. Recall the inverse hyperbolic function identity: $$\operatorname{arcosh}(z) = \ln(z + \sqrt{z^2$ - 1})$$ and the derivative of $$\operatorname{arcosh}(x) is$$ $$\frac{1}{\sqrt{x^2$ - 1}}$$, but here the integrand involves $$\sqrt{1 - 16x^2$}$$, so consider the substitution $$x = \frac{1}{4} \sin 	heta or $$relate it to $$\operatorname{arcosh} or$$ $$\operatorname{arcsinh}$$ forms. Use the substitution $$x = \frac{1}{4} \sin 	heta to $$rewrite the integral in terms of $$	heta$$, then simplify the integral to a form involving $$\csc 	heta or$$ $$\cot 	heta$$, which can be integrated to inverse hyperbolic functions. After integrating with respect to $$	heta$$, back-substitute to express the answer in terms of $$x$$, and then evaluate the definite integral by plugging in the limits $$x = \frac{1}{5}$$ and $$x = \frac{3}{13}.$$

Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))

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Concetti chiave

Inverse Hyperbolic Functions

Integration Techniques for Radical Expressions

Definite Integration with Substitution