Why Exponential Antiderivatives Need Care
Exponential functions are unusual among familiar functions because their derivatives are proportional to the original function. That makes their antiderivatives closely related to themselves, but the proportionality factor matters.
For the natural exponential \(e^x\), the derivative is \(e^x\), so its antiderivative has the same form. For another positive base, such as \(5\), differentiation introduces a factor of \(\ln 5\). To reverse that derivative, an antiderivative must compensate by dividing by \(\ln 5\).
This distinction is useful whenever an exponential function describes a quantity whose rate of change is proportional to its current value. Finding an antiderivative recovers a function with that exponential rate, while differentiation provides a direct check that the factor is correct.
The Exponential Derivative Rule
For \(a>0\), the real exponential function with base \(a\) can be written as \(a^x=e^{x\ln a}\). By the chain rule and the derivative of \(e^x\),
Here \(a\) is a fixed positive number, so \(\ln a\) is constant with respect to \(x\). In the special case \(a=e\), \(\ln e=1\), and therefore \(\frac{d}{dx}e^x=e^x\).
Deriving the Antiderivative Formula
Suppose \(a>0\) and \(a\ne1\). Since \(\ln a\ne0\), we can look for an antiderivative of \(a^x\) in the form \(K a^x\), where \(K\) is a constant. Differentiating gives
For this derivative to equal \(a^x\), the constant must satisfy \(K\ln a=1\). Thus \(K=1/\ln a\). The derivation also shows why dividing by the logarithm of the base is necessary.
When the base is \(e\), its logarithm is one, so this formula reduces to the particularly simple result below.
The constant \(C\) appears because adding a constant to a function does not change its derivative. In fact, on an interval, any two antiderivatives of the same function differ by a constant: their difference has derivative zero, and the Mean Value Theorem implies that a function with zero derivative throughout an interval is constant there.
Worked Examples
Worked Example: Integrating the Natural Exponential
Find \(\int e^x\,dx\), then check the result.
The derivative of \(e^x\) is \(e^x\), so an antiderivative is
To check, differentiate the proposed antiderivative:
The derivative matches the integrand, so the result is correct. The constant \(C\) represents all possible vertical shifts of this antiderivative.
Worked Example: Integrating an Exponential with Base 5
Find \(\int 5^x\,dx\), then verify the base-dependent factor.
The base is positive and is not one, so use the formula with denominator \(\ln 5\):
Differentiate the nonconstant part. Since \(\ln 5\) is a constant,
The factors \(\ln 5\) cancel, leaving exactly the original integrand. This confirms that the denominator is required.
Worked Example: A Base Between Zero and One
Find an antiderivative of \(6(1/3)^x\) and check it.
The base \(1/3\) is positive and not one, so the exponential formula applies. By the logarithm rule \(\ln(1/3)=-\ln 3\),
Now differentiate the proposed expression:
The derivative equals the integrand. The negative coefficient in the antiderivative is correct because \(\ln(1/3)\) is negative.
Worked Example: The Exceptional Base 1
Find \(\int 1^x\,dx\). The general formula with denominator \(\ln a\) does not apply at \(a=1\), because \(\ln 1=0\).
Since \(1^x=1\) for every real \(x\), an antiderivative is
Check by differentiation:
Thus base one has a simple antiderivative, but it must be handled separately from the formula for \(a\ne1\).
Why the Formula Works for Every Allowed Base
The base-dependent formula can be verified directly, without guessing. For \(a>0\), \(a\ne1\), differentiate the proposed expression:
This check establishes that the expression is an antiderivative. To see that it gives the full family, let \(F\) be any other antiderivative of \(a^x\) on the same interval. The difference between \(F\) and the displayed antiderivative has derivative
A differentiable function with derivative zero on an interval is constant there, by the Mean Value Theorem. Hence \(F(x)-a^x/\ln a=C\) for some constant \(C\), which gives \(F(x)=a^x/\ln a+C\). The same reasoning applies to \(e^x\), whose derivative is itself.
A Common Mistake: Multiplying by \(\ln a\)
A frequent error is to place \(\ln a\) in the numerator and propose \(a^x\ln a\) as an antiderivative of \(a^x\). Differentiating that expression shows why the placement matters:
This generally is not \(a^x\). It equals \(a^x\) only when \((\ln a)^2=1\), which occurs for \(a=e\) or \(a=1/e\). For example, at base \(2\), the incorrect proposal gives
because \((\ln 2)^2\ne1\). The corrected antiderivative is \(2^x/\ln 2+C\), and its derivative is
The exceptional bases do not change the general formula: for \(e\) and \(1/e\), dividing by \(\ln a\) still gives a valid antiderivative. The expression with \(\ln a\) in the numerator happens to work at those bases because its extra factor squares to one.
Choosing the Correct Case
Before applying an exponential antiderivative formula, identify the base. The cases can be summarized as follows:
| Integrand | Antiderivative | Condition |
|---|---|---|
| \(e^x\) | \(e^x+C\) | Natural exponential |
| \(a^x\) | \(\frac{a^x}{\ln a}+C\) | \(a>0\), \(a\ne1\) |
| \(1^x=1\) | \(x+C\) | Base equals one |
If an integrand is a constant multiple of one of these functions, that constant multiple remains in the answer. For example, differentiating \(k a^x/\ln a\), where \(k\) is constant and \(a>0\), \(a\ne1\), gives
This is simply the same derivative check with the fixed coefficient carried through. For any proposed antiderivative, differentiating it and comparing the result with the original integrand is the most reliable way to catch a missing or misplaced factor.
Check Your Understanding
Use the derivative rule for \(a^x\) to solve each problem, and check your answer by differentiation.
- Find an antiderivative of \(e^x\).
- Find an antiderivative of \(7^x\). What factor involving the base is required?
- Find an antiderivative of \((1/5)^x\). Explain the sign of its coefficient.
- Why does the formula \(\frac{a^x}{\ln a}+C\) require \(a\ne1\), and what is an antiderivative when \(a=1\)?
- Differentiate \(a^x\ln a\). For which positive bases does this derivative equal \(a^x\)?
