Exponential and Logarithmic Integration

This section covers:

Exponential and Logarithmic Differentiation and Integration have a lot of practical applications and are handled a little differently than we are used to. For a review of these functions, visit the Exponential Functions section and the Logarithmic Functions section.

Note that Exponential and Logarithmic Differentiation is covered here.

Introduction to Exponential and Logarithmic Differentiation and Integration

Before getting started, here is a table of the most common Exponential and Logarithmic formulas for Differentiation and Integration:

Exponential and Logarithmic Derivatives Exponential and Logarithmic Integrals Examples
\(\displaystyle \frac{d}{{dx}}\left( {\ln u} \right)=\frac{{{u}’}}{u}\) \(\displaystyle \int{{\frac{{{u}’}}{u}}}\,dx=\ln \left| u \right|\,+C\)

     \(\displaystyle \color{#800000}{{\frac{{d\left[ {\ln \left( {{{x}^{5}}-3} \right)} \right]}}{{dx}}}}=\frac{{{u}’}}{u}=\frac{{5{{x}^{4}}}}{{{{x}^{5}}-3}}\,\,\,\,\,\,\,\text{(}u={{x}^{5}}-3)\)


\(\displaystyle \begin{align}u&={{x}^{5}}-3\\du&=5{{x}^{4}}\,dx\\dx&=\frac{{du}}{{5{{x}^{4}}}}\end{align}\)          \(\require{cancel}  \begin{align}\color{#800000}{{\int{{\frac{{5{{x}^{4}}}}{{{{x}^{5}}-3}}}}\,dx}}&=\int{{\frac{{du}}{u}}}=\ln \left| u \right|+C\\&=\ln \left| {{{x}^{5}}-3} \right|+C\end{align}\)

\(\displaystyle \frac{d}{{dx}}\left( {{{{\log }}_{a}}u} \right)=\frac{{{u}’}}{{u\left( {\ln \,a} \right)}}\)

\(\displaystyle \int{{\frac{{{u}’}}{{u\ln a}}}}\,dx={{\log }_{a}}\left| u \right|+C\)

(Not a typical integration problem)

\(\displaystyle \color{#800000}{{\frac{d}{{dx}}\left[ {\log \left( {4x-1} \right)} \right]}}=\frac{d}{{dx}}\left[ {{{{\log }}_{{10}}}\left( {4x-1} \right)} \right]=\frac{4}{{\left( {4x-1} \right)\ln 10}}\)
\(\displaystyle \frac{d}{{dx}}\left( {{{e}^{u}}} \right)={{e}^{u}}{u}’\) \(\int{{{{e}^{u}}}}={{e}^{u}}+C\)

\(\displaystyle \color{#800000}{{\frac{{d\left( {{{e}^{{3x}}}} \right)}}{{dx}}}}={{e}^{{3x}}}\cdot 3=3{{e}^{{3x}}}\)


\(\begin{array}{l}u=3x\\du=3\,dx\end{array}\)             \(\displaystyle \color{#800000}{{\int{{3{{e}^{{3x}}}}}dx}}=3\int{{{{e}^{{3x}}}dx}}={{e}^{{3x}}}+C\)

\(\displaystyle \frac{d}{{dx}}\left( {{{a}^{u}}} \right)=\left( {\ln \,a} \right){{a}^{u}}{u}’\) \(\displaystyle \int{{{{a}^{u}}}}du=\left( {\frac{1}{{\ln a}}} \right){{a}^{u}}+C\)        \(\displaystyle \color{#800000}{{\frac{d}{{dx}}\left( {{{4}^{{3{{x}^{3}}}}}} \right)}}=\ln 4\left( {{{4}^{{3{{x}^{3}}}}}} \right)\left( {9{{x}^{2}}} \right)\)


\(\displaystyle \begin{align}u&=3{{x}^{3}}\\du&=9{{x}^{2}}\,dx\\dx&=\frac{{du}}{{9{{x}^{2}}}}\end{align}\)            \(\displaystyle \begin{array}{l}\color{#800000}{{\int{{\ln 4\left( {{{4}^{{3{{x}^{3}}}}}} \right)}}\left( {9{{x}^{2}}} \right)dx}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,=\ln 4\int{{\left( {{{4}^{u}}} \right)}}\left( {\cancel{{9{{x}^{2}}}}} \right)\frac{{du}}{{\cancel{{9{{x}^{2}}}}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,=\ln 4\cdot \frac{1}{{\ln 4}}\cdot {{4}^{u}}+C={{4}^{{3{{x}^{3}}}}}+C\end{array}\)

\(\displaystyle \frac{d}{{dx}}\left[ {f{{{\left( x \right)}}^{{g\left( x \right)}}}} \right]\)


When we have a variable both in the base and the exponent, or if the function is really complicated, it’s best to take ln of both sides to take derivative, and use implicit integration. Then substitute \(y\) back in at the end.

Find Derivative: \(\displaystyle y=3{{x}^{x}}\)


\(\displaystyle \begin{align}\ln y&=\ln {{\left( {3x} \right)}^{x}};\ln y=x\ln \left( {3x} \right)\\\frac{1}{y}\left( {\frac{{dy}}{{dx}}} \right)&=x\cdot \frac{3}{x}+\ln \left( {3x} \right)\cdot 1\,\,\,\,\text{(product rule)}\\\,\frac{{dy}}{{dx}}&=\left[ {3+\ln \left( {3x} \right)} \right]\cdot y=3{{x}^{x}}\left[ {3+\ln \left( {3x} \right)} \right]\end{align}\)

Integrals of the Six Basic Trig Functions

\(\displaystyle \begin{array}{c}\int{{\sin u\,du=-\cos u\,\,+\,\,C}}\\\int{{\tan u\,du=-\ln \left| {\cos u} \right|\,\,+\,\,C\,}}\\\int{{\sec u\,du=\ln \left| {\sec u+\tan u} \right|\,\,+\,\,C}}\,\\\int{{\cos u\,du=\sin u\,\,+\,\,C}}\\\int{{\cot u\,du=\ln \left| {\sin u} \right|\,\,+\,\,C}}\\\int{{\csc u\,du=-\ln \left| {\csc u+\cot u} \right|\,\,+\,\,C}}\end{array}\)

\(\begin{align}u&=4\theta \\du&=4\,d\theta \end{align}\)          \(\begin{align}\color{#800000}{{\int{{\csc \left( {4\theta } \right)}}\,d\theta }}&=\frac{1}{4}\int{{\csc u\,du}}\\&=-\frac{1}{4}\ln \left| {\csc \left( {4x} \right)+\cot \left( {4x} \right)} \right|+C\end{align}\)


Actually, when we take the integrals of exponential and logarithmic functions, we’ll be using a lot of U-Sub Integration, so you may want to review it.

Review of Logarithms

When we learned the Power Rule for Integration here in the Antiderivatives and Integration section, we noticed that if \(n=-1\), the rule doesn’t apply: \(\displaystyle \int{{{{x}^{n}}}}dx=\frac{{{{x}^{{n+1}}}}}{{n+1}}\,+C,\,\,n\ne 1\). So when we try to integrate a function like \(\displaystyle f\left( x \right)=\frac{1}{x}={{x}^{{-1}}}\), we have to do something “special”; namely learn that this integral is \(\ln \left( x \right)\).

Remember that \(\ln x\) is the same as \({{\log }_{e}}x\), where \(e\approx 2.718\) (“\(e\)” is Euler’s Number). A log is the exponent raised to the base power (\(a\)) to get the argument (\(x\)) of the log (if “\(a\)” is missing, we assume it’s 10).

Here are some logarithmic properties that we learned here in the Logarithmic Functions section; note we could use \({{\log }_{a}}x\) instead of \(\ln x\).

Logarithmic Properties


\(\begin{array}{c}\ln \left( {{{a}^{n}}} \right)=n\ln a\\\ln \left( {ab} \right)=\ln \left( a \right)+\ln \left( b \right)\\\ln \left( {\frac{a}{b}} \right)=\ln \left( a \right)-\ln \left( b \right)\\\ln \left( 1 \right)=0\\\ln \left( {{{e}^{x}}} \right)=x\end{array}\)               \(\displaystyle \begin{align}\text{Example:}\\\ln {{\left( {\frac{{{{x}^{4}}y}}{{\sqrt{z}}}} \right)}^{2}}&=2\ln \left( {\frac{{{{x}^{4}}y}}{{{{z}^{{\frac{1}{2}}}}}}} \right)=2\left[ {\ln \left( {{{x}^{4}}} \right)+\ln \left( y \right)-\ln \left( {{{z}^{{\frac{1}{2}}}}} \right)} \right]\\&=2\left[ {4\ln \left( x \right)+\ln \left( y \right)-\frac{1}{2}\ln \left( z \right)} \right]\\&=8\ln \left( x \right)+2\ln \left( y \right)-\ln \left( z \right)\end{align}\)

The Log Rule for Integration

We learned that the differentiation rule for log functions is \(\displaystyle \frac{d}{{dx}}\left[ {\ln u} \right]du=\frac{{{u}’}}{u}\).


From this, we can get the Log Rules for Integration; you’ll probably just want to memorize these. Remember that | | is the absolute value function, which means always take the positive of what’s inside. The reason we use an absolute value is that the natural logarithm function is only defined for \(x>0\).

Log Rule For Integration


Most of these problems involve U-Sub and some require doing polynomial long division before integrating when the degree (largest exponent of all the terms) of the numerator is greater than the degree of the denominator.

Let’s do some Logarithmic Integration problems:

Log Integration Problems

Integrals of Trigonometric Functions using “ln”

We learned integrals of some of the trig functions here in the Antiderivatives and Indefinite Integration section, but now that we know some log rules, we’ll introduce the rest of the trig integrals. The new trig integrals may be proved by using the log integration rules, but you’ll probably just want to memorize these:


Integrals of Trig Functions

Here are some Trig Integration problems; notice that sometimes we can’t use the above equations, but have to work with the log integral rules:

Trig Integration Problems

Integrals of eu and au

Now let’s take the integrals of the exponential functions \({{e}^{u}}\) and \({{a}^{u}}\), which mainly involves U-sub. When we take the integral of a base other than \(e\), we can either convert the function to base \(e\) using the formula \({{a}^{x}}={{e}^{{\left( {\ln a} \right)x}}}\) (since \(\displaystyle {{e}^{{\left( {\ln a} \right)x}}}={{\left( {{{e}^{{\left( {\ln a} \right)}}}} \right)}^{x}}={{a}^{x}}\)), or remember the formula below:

Exponential Integrals

Let’s do some Exponential Integration problems. You can also differentiate back to make sure you got the right answer!

First, the eu integration problems. Note that if there are multiple instances of \({{e}^{x}}\) in the problem, we usually include \({{e}^{x}}\) in the \(u\) part of the problem; otherwise we don’t.

E To The X Integration Problems

And now, the au  integration problems:

A To The X Integration Problem Understand these rules and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Exponential Growth Using Calculus – you’re ready!