5.6 Integrals involving exponential and logarithmic functions

Exponential and logarithmic functions are used to model population growth, cell growth, and financial growth, as well as depreciation, radioactive decay, and resource consumption, to name only a few applications. In this section, we explore integration involving exponential and logarithmic functions.

Integrals of exponential functions

The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, $y=e^x,$ is its own derivative and its own integral.

A common mistake when dealing with exponential expressions is treating the exponent on e the same way we treat exponents in polynomial expressions. We cannot use the power rule for the exponent on e . This can be especially confusing when we have both exponentials and polynomials in the same expression, as in the previous checkpoint. In these cases, we should always double-check to make sure we’re using the right rules for the functions we’re integrating.

Square root of an exponential function

Find the antiderivative of the exponential function $e^x\sqrt{1+e^x}.$

First rewrite the problem using a rational exponent:

${\displaystyle \int e^x\sqrt{1+e^x}}dx={\displaystyle \int e^x{\left(1+e^x\right)}^{1\text{/}2}dx}.$

Using substitution, choose $u=1+e^x.u=1+e^x.$ Then, $du=e^{x}dx.$ We have ( [link] )

${\displaystyle \int e^x{\left(1+e^x\right)}^{1\text{/}2}dx={\displaystyle \int u^{1\text{/}2}du}}.$

Then

${\displaystyle \int u^{1\text{/}2}du=\frac{u^{3\text{/}2}}{3\text{/}2}+C}=\frac{2}{3}{u}^{3\text{/}2}+C=\frac{2}{3}{\left(1+e^x\right)}^{3\text{/}2}+C.$

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As mentioned at the beginning of this section, exponential functions are used in many real-life applications. The number e is often associated with compounded or accelerating growth, as we have seen in earlier sections about the derivative. Although the derivative represents a rate of change or a growth rate, the integral represents the total change or the total growth. Let’s look at an example in which integration of an exponential function solves a common business application.