Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
67. ∫(from 0 to 2√3)dx/√(4+x²)

검증된 단계별 안내

Recognize that the integral \( \int \frac{dx}{\sqrt{a^2 + x^2}} \) can be expressed in terms of the inverse hyperbolic sine function, \( \sinh^{-1}(x/a) \), where \( a \) is a constant.

Identify the constant \( a \) in the integral. Here, the integral is \( \int_0^{2\sqrt{3}} \frac{dx}{\sqrt{4 + x^2}} \), so \( a^2 = 4 \) which means \( a = 2 \).

Rewrite the integral using the formula: \( \int \frac{dx}{\sqrt{a^2 + x^2}} = \sinh^{-1}\left( \frac{x}{a} \right) + C \). For the definite integral, evaluate \( \sinh^{-1}\left( \frac{x}{2} \right) \) from 0 to \( 2\sqrt{3} \).

Set up the evaluation: compute \( \sinh^{-1}\left( \frac{2\sqrt{3}}{2} \right) - \sinh^{-1}\left( \frac{0}{2} \right) \). Simplify the arguments inside the inverse hyperbolic sine functions.

Recall that \( \sinh^{-1}(y) = \ln\left(y + \sqrt{y^2 + 1}\right) \) if you want to express the answer in logarithmic form, but since the problem asks for inverse hyperbolic functions, leave the answer in terms of \( \sinh^{-1} \).

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Inverse Hyperbolic Functions

Inverse hyperbolic functions, such as sinh⁻¹(x) and cosh⁻¹(x), are the inverses of hyperbolic sine and cosine functions. They often appear in integrals involving expressions like √(a² + x²). Recognizing these forms allows rewriting integrals in terms of inverse hyperbolic functions for simpler evaluation.

Integration of Rational Functions Involving Square Roots

Integrals containing expressions like 1/√(a² + x²) are common and can be solved using substitution or by recalling standard integral formulas. These integrals typically result in inverse hyperbolic functions, making it essential to identify the pattern to apply the correct formula efficiently.

Definite Integration and Evaluation of Limits

Definite integrals require evaluating the antiderivative at the upper and lower limits. After finding the integral in terms of inverse hyperbolic functions, substituting the limits correctly and simplifying the result is crucial to obtain the final numerical value.

Evaluate the integrals in Exercises 67–74 in terms ofa. inverse hyperbolic functions.67. ∫(from 0 to 2√3)dx/√(4+x²)