Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

문제 7.3.37

7장, 문제 7.3.37

37–56. Integrals Evaluate each integral.
∫ cosh 2x dx

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Recall the definition of the hyperbolic cosine function: \(\cosh x = \frac{e^x + e^{-x}}{2}\).

Use the double angle identity for hyperbolic cosine: \(\cosh 2x = 2\cosh^2 x - 1\) or express \(\cosh 2x\) in terms of exponentials as \(\frac{e^{2x} + e^{-2x}}{2}\).

Rewrite the integral using the exponential form: \(\int \cosh 2x \, dx = \int \frac{e^{2x} + e^{-2x}}{2} \, dx\).

Split the integral into two simpler integrals: \(\frac{1}{2} \int e^{2x} \, dx + \frac{1}{2} \int e^{-2x} \, dx\).

Integrate each term using the formula \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\), then combine the results and add the constant of integration.

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Hyperbolic Functions

Hyperbolic functions, such as cosh(x) and sinh(x), are analogs of trigonometric functions but based on exponential functions. Specifically, cosh(x) = (e^x + e^(-x))/2. Understanding their definitions and properties is essential for integrating expressions involving cosh.

Integration of Exponential Functions

Since hyperbolic functions are expressed in terms of exponentials, integrating cosh(2x) involves integrating exponential functions like e^(2x). The integral of e^(ax) with respect to x is (1/a)e^(ax) + C, which helps in solving the integral of cosh(2x).

Use of Substitution in Integration

When integrating functions like cosh(2x), substitution is useful to simplify the integral. Letting u = 2x transforms the integral into a simpler form, allowing the use of standard integration rules and then substituting back to the original variable.

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tanh(−x) = −tanh x

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22–36. Derivatives Find the derivatives of the following functions.

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