Integration of Exponential, Logarithmic, and Hyperbolic Functions: Cheat Sheet

Download the Spanish version here.

Essential Concepts

Integrals, Exponential Functions, and Logarithms

The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.
The cornerstone of the development is the definition of the natural logarithm in terms of an integral.
The function [latex]{e}^{x}[/latex] is then defined as the inverse of the natural logarithm.
General exponential functions are defined in terms of [latex]{e}^{x},[/latex] and the corresponding inverse functions are general logarithms.
Familiar properties of logarithms and exponents still hold in this more rigorous context.

Exponential Growth and Decay

Exponential growth and exponential decay are two of the most common applications of exponential functions.
Systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}.[/latex]
In exponential growth, the rate of growth is proportional to the quantity present. In other words, [latex]{y}^{\prime }=ky.[/latex]
Systems that exhibit exponential growth have a constant doubling time, which is given by [latex](\text{ln}2)\text{/}k.[/latex]
Systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\text{−}kt}.[/latex]
Systems that exhibit exponential decay have a constant half-life, which is given by [latex](\text{ln}2)\text{/}k.[/latex]

Calculus of the Hyperbolic Functions

Hyperbolic functions are defined in terms of exponential functions.
Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.
With appropriate range restrictions, the hyperbolic functions all have inverses.
Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.
The most common physical applications of hyperbolic functions are calculations involving catenaries.

catenary: a curve in the shape of the function [latex]y=a\text{cosh}(x\text{/}a)[/latex] is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary

doubling time: if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by [latex]\frac{(\text{ln}2)}{k}[/latex]

exponential decay: systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\text{−}kt}[/latex]

exponential growth: systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[/latex]

half-life: if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by [latex]\frac{(\text{ln}2)}{k}[/latex]

The Natural Logarithm as an Integral

Properties of the Exponential Function

Visualize [latex]e[/latex] as the [latex]x[/latex]-coordinate where area under [latex]\frac{1}{t}[/latex] curve equals [latex]1[/latex]
Practice using exponential properties to manipulate expressions
Connect differentiation and integration of exponential functions

General Logarithmic and Exponential Functions

Practice converting between general exponential and logarithmic forms.
Memorize the key derivative and integral formulas for [latex]a^x[/latex] and [latex]\log_a x[/latex].
Remember that all logarithmic functions are constant multiples of each other.
When differentiating or integrating expressions with general bases, convert to natural logarithms or exponentials first if needed.
Pay attention to the chain rule when dealing with composite functions involving exponentials or logarithms.

Exponential Growth Model

Understand the significance of the growth constant [latex]k[/latex] in determining the rate of growth.
When solving exponential growth problems, use logarithms to isolate the variable of interest.
For compound interest problems, pay attention to whether the interest is compounded continuously or at discrete intervals.
Remember that the doubling time is constant in exponential growth, regardless of the initial quantity.

Exponential Decay Model

Understand the significance of the decay constant [latex]k[/latex] in determining the rate of decay.
When solving exponential decay problems, use logarithms to isolate the variable of interest.
For Newton’s Law of Cooling problems, pay attention to the ambient temperature and its effect on the decay rate.
Remember that the half-life is constant in exponential decay, regardless of the initial quantity.

Derivatives and Integrals of the Hyperbolic Functions

Memorize the definitions of [latex]\sinh[/latex] and [latex]\cosh[/latex] in terms of exponentials.
Practice deriving the derivatives of hyperbolic functions from their definitions.
Notice the patterns in the derivative formulas and compare them with trigonometric function derivatives.
When integrating, look for opportunities to use u-substitution, especially with composites of hyperbolic functions.
Remember that [latex]\cosh x[/latex] is always positive, which can simplify some integrals (e.g., when dealing with absolute value).
When solving problems, consider whether using hyperbolic function identities might simplify the expression.

Calculus of Inverse Hyperbolic Functions

Memorize the domains and ranges of inverse hyperbolic functions.
Notice the similarities between derivatives of inverse hyperbolic and inverse trigonometric functions.
When integrating, pay attention to the form of the integrand to identify which inverse hyperbolic function to use.
Remember that [latex]\tanh^{-1} x[/latex] and [latex]\coth^{-1} x[/latex] have the same derivative, so context is important when integrating [latex]\frac{1}{1-x^2}[/latex].
When solving problems, consider whether using hyperbolic function identities might simplify the expression.

Applications of Hyperbolic Functions

Understand the physical meaning of the parameters in the catenary equation [latex]y = a \cosh(\frac{x}{a})[/latex].
Practice sketching catenary curves for different values of ‘[latex]a[/latex]‘ to understand how it affects the shape.
When solving catenary problems, pay attention to the units of measurement given and ensure your answer uses the correct units.
Practice deriving the derivative of catenary functions, as this is often needed in arc length calculations.