Variations on the substitution method Evaluate the following integrals.                                                                                                        
                                                                                                                                                                    
 ∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍

Verified step by step guidance

1. 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.
2. 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 \(\frac{2 \sinh x}{2 \cosh x} = \frac{\sinh x}{\cosh x} = \tanh x\).
3. Rewrite the integral as \(\int \tanh x \, d\!x\) to simplify the problem.
4. Recall that the derivative of \(\ln(\cosh x)\) is \(\tanh x\), so the integral of \(\tanh x\) with respect to \(x\) is \(\ln|\cosh x| + C\).
5. 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.

Key Concepts

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.

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.

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.