# Calculus: Integrals III

## Exponential Functions

The integral of the exponential function is as easy as the derivative:

$$\displaystyle\int e^x \, dx = e^x + C$$

For a general exponential function, the integral is nearly as straightforward:

$$\displaystyle\int a^{x} \, dx = \dfrac{1}{\ln(a)}a^{x} + C$$

Later, you will be able to derive the second formula from the first one in the section on $u$-substitution.

## Problems

1. Evaluate: $\displaystyle\int e^{x+1} \, dx$

$\displaystyle\int e^{x+1} \, dx = \displaystyle\int e^xe^1 \, dx \\ \displaystyle\int e^{x+1} \, dx = e\displaystyle\int e^x \, dx \\ \displaystyle\int e^{x+1} \, dx = e\left( e^x + c \right) \\ \displaystyle\int e^{x+1} \, dx = e^{x+1} + ec \\ \displaystyle\int e^{x+1} \, dx = e^{x+1} + c \\$

2. Evaluate: $\displaystyle\int 2^{x/ \ln(2)} \,dx$

$\displaystyle\int 2^{x/ \ln(2)} \,dx = \displaystyle\int e^{\ln(2)x/ \ln(2)} \, dx \\ \displaystyle\int 2^{x/ \ln(2)} \,dx = \displaystyle\int e^{x} \, dx \\ \displaystyle\int 2^{x/ \ln(2)} \,dx = e^x + c \\$

3. Evaluate: $\displaystyle\int 4^x \, dx$

$\displaystyle\int 4^x \, dx = \dfrac{1}{\ln(4)}4^x + c$

4. Evaluate: $\displaystyle\int 3^{3x} \, dx$

$\displaystyle\int 3^{3x} \, dx = \displaystyle\int 27^x \, dx \\ \displaystyle\int 3^{3x} \, dx = \dfrac{1}{\ln(27)}27^x + c \\$

5. Evaluate: $\displaystyle\int \pi^x \, dx$

$\displaystyle\int \pi^x \, dx = \dfrac{1}{\ln(\pi)}\pi^x + c$
6. Evaluate $\displaystyle\int \alpha^{2x} \,dx$

$\displaystyle\int \alpha^{2x} \,dx = \displaystyle\int (\alpha^2)^x \, dx \\ \displaystyle\int \alpha^{2x} \,dx = \dfrac{1}{\ln(\alpha^2)}(\alpha^2)^x + c \\ \displaystyle\int \alpha^{2x} \,dx = \dfrac{1}{2\ln(\alpha)}\alpha^{2x} + c \\$

7. Evaluate $\displaystyle\int e^{\ln(4)x} \, dx$

$\displaystyle\int e^{\ln(4)x} \, dx = \displaystyle\int 4^{x} \, dx \\ \displaystyle\int e^{\ln(4)x} \, dx = \dfrac{1}{\ln(4)}4^{x} + c\\$

8. Evaluate: $\displaystyle\int 9^{x/2} \, dx$

$\displaystyle\int 9^{x/2} \, dx = \displaystyle\int 3^x \, dx \\ \displaystyle\int 9^{x/2} \, dx = \dfrac{1}{\ln(3)}3^x + c \\$
9. Evaluate: $\displaystyle\int e^{-x} \, dx$

$\displaystyle\int e^{-x} \, dx = \displaystyle\int \left(\dfrac{1}{e}\right)^{x} \, dx\\ \displaystyle\int e^{-x} \, dx = \dfrac{1}{\ln(1/e)}\left(\dfrac{1}{e}\right)^{x} + c \\ \displaystyle\int e^{-x} \, dx = -e^{-x} + c \\$

10. Evaluate $\displaystyle\int 2^{-x} \, dx$

$\displaystyle\int 2^{-x} \, dx = \displaystyle\int \left(\dfrac{1}{2}\right)^{x} \, dx \\ \displaystyle\int 2^{-x} \, dx = \dfrac{1}{\ln(1/2)}\left(\dfrac{1}{2}\right)^{x} + c \\ \displaystyle\int 2^{-x} \, dx = -\dfrac{1}{\ln(2)}2^{-x} + c \\$