Variations on the substitution method Evaluate the following integrals.

∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍

Guida verificata passo dopo passo

Recognize that the integral is of the form \(\int \frac{e^x - e^{-x}}{e^x + e^{-x}} \, d\!x\), which suggests a substitution involving the denominator or a related function.

Recall the hyperbolic functions: \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\). Notice that the numerator is \(2 \sinh x\) and the denominator is \(2 \cosh x\), so the integrand simplifies to \(\frac{2 \sinh x}{2 \cosh x} = \frac{\sinh x}{\cosh x} = \tanh x\).

Rewrite the integral as \(\int \tanh x \, d\!x\) to simplify the problem.

Recall that the derivative of \(\ln(\cosh x)\) is \(\tanh x\), so the integral of \(\tanh x\) with respect to \(x\) is \(\ln|\cosh x| + C\).

Write the final integral expression as \(\int \frac{e^x - e^{-x}}{e^x + e^{-x}} \, d\!x = \ln|\cosh x| + C\), where \(C\) is the constant of integration.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform a complicated integral into a basic form. It involves identifying a part of the integrand as a new variable, differentiating it, and rewriting the integral in terms of this variable. This technique is especially useful when the integral contains composite functions.

Video consigliato:

07:33

Euler's Method

Hyperbolic Functions and Their Properties

Expressions involving eˣ and e⁻ˣ often relate to hyperbolic functions such as sinh(x) and cosh(x). Recognizing these can simplify integration since sinh(x) = (eˣ - e⁻ˣ)/2 and cosh(x) = (eˣ + e⁻ˣ)/2. Using these identities helps rewrite the integral in a more manageable form.

Video consigliato:

Percorso guidato

06:21

Properties of Functions

Integration of Rational Functions

Integrals involving ratios of functions, like (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ), require understanding how to manipulate and simplify rational expressions. This often involves algebraic simplification or substitution to reduce the integral to a standard form, enabling straightforward integration.

Video consigliato: