37–56. Integrals Evaluate each integral.

∫₀ ˡⁿ ² tanh x dx

검증된 단계별 안내

Recognize that the integral to evaluate is \(\int_0^{\ln 2} \tanh x \, dx\), where \(\tanh x = \frac{\sinh x}{\cosh x}\).

Recall the definition of hyperbolic tangent: \(\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}\), but it is often easier to work with the derivative of \(\ln(\cosh x)\) since \(\frac{d}{dx} \ln(\cosh x) = \tanh x\).

Use the fact that \(\frac{d}{dx} \ln(\cosh x) = \tanh x\) to rewrite the integral as \(\int_0^{\ln 2} \tanh x \, dx = \left[ \ln(\cosh x) \right]_0^{\ln 2}\).

Evaluate the expression \(\ln(\cosh x)\) at the upper limit \(x = \ln 2\) and the lower limit \(x = 0\) separately.

Subtract the value at the lower limit from the value at the upper limit to find the value of the definite integral: \(\ln(\cosh(\ln 2)) - \ln(\cosh(0))\).

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Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Hyperbolic Functions and Their Properties

Hyperbolic functions like tanh(x) are analogs of trigonometric functions but based on exponential functions. The function tanh(x) = (e^x - e^{-x}) / (e^x + e^{-x}) is continuous and differentiable, with known derivatives and integrals that simplify integration tasks.

Integration Techniques for Hyperbolic Functions

Integrating hyperbolic functions often involves recognizing standard integral forms or using substitution. For tanh(x), the integral can be expressed in terms of logarithmic functions, since d/dx [ln(cosh x)] = tanh x, which helps in evaluating definite integrals efficiently.

37–56. Integrals Evaluate each integral.∫₀ ˡⁿ ² tanh x dx