Evaluate the integral: integral (3^x - 4^x) dx
Evaluate the integral: $$ \int \left(3^x - 4^x\right) dx. $$
Explanation of Method
For a general exponential base $a>0,\ a\neq 1$, the antiderivative is $\int a^x\,dx = \dfrac{a^x}{\ln a} + C$ — a direct generalization of $\int e^x\,dx = e^x + C$. When the integrand is a sum or difference of such terms, apply the formula to each base separately and combine.
- Split the integral into one term per exponential base.
- Apply the general exponential integration formula to each term.
- Combine the results and add the constant of integration.
Step 1 · Recall the integral of an exponential function
The integral can be expressed as:
$$ \int \left(3^x - 4^x\right) dx = \int 3^x\,dx - \int 4^x\,dx. $$
The general formula for the integral of $a^x$, where $a>0$ and $a\neq 1$, is:
$$ \int a^x\,dx = \frac{a^x}{\ln(a)} + C. $$
Step 2 · Apply the formula to each term
First term: $\displaystyle \int 3^x\,dx$. Using the formula, the integral is:
$$ \int 3^x\,dx = \frac{3^x}{\ln(3)}. $$
Second term: $\displaystyle \int 4^x\,dx$. Similarly, using the formula, the integral is:
$$ \int 4^x\,dx = \frac{4^x}{\ln(4)}. $$
Step 3 · Combine the results
Substituting back into the original integral, we get:
$$ \int \left(3^x - 4^x\right) dx = \frac{3^x}{\ln(3)} - \frac{4^x}{\ln(4)} + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Base $a$ | Antiderivative |
|---|---|---|
| $3^x$ | $3$ | $\dfrac{3^x}{\ln 3}$ |
| $-4^x$ | $4$ | $-\dfrac{4^x}{\ln 4}$ |
Additional Notes
- The formula $\int a^x\,dx = \dfrac{a^x}{\ln a}+C$ reduces to $\int e^x\,dx = e^x+C$ when $a=e$, since $\ln e = 1$.
- The integrand is defined for all real $x$, since exponential functions with positive base are defined everywhere.
- The result can be checked by differentiating $\dfrac{3^x}{\ln 3}-\dfrac{4^x}{\ln 4}$ and confirming it returns the original integrand.
Result
The solution to the integral is: