Integration of Exponential Functions

THE INTEGRATION OF EXPONENTIAL FUNCTIONS

The following problems involve the integration of exponential functions. We will assume knowledge of the following well-known differentiation formulas :

$\displaystyle{ D \{ e^x \} = e^x }$ ,

where $e \approx 2.71828$ , and

$\displaystyle{ D \{ a^x \} = a^x \ln a }$ ,

where a is any positive constant not equal to 1 and $\ln a$ is the natural (base e) logarithm of a . These formulas lead immediately to the following indefinite integrals :

$\displaystyle{ \int e^x \,dx } = e^x + C$
$\displaystyle{ \int a^x \,dx } = { a^x \over \ln a } + C$

As you do the following problems, remember these three general rules for integration :

$\displaystyle{ \int x^n \,dx } = { x^{n+1} \over n+1 } + C$ ,

where n is any constant not equal to -1,

$\displaystyle{ \int k f(x) \,dx } = k \displaystyle{ \int f(x) \,dx }$ ,

where k is any constant, and

$\displaystyle{ \int ( f(x) \pm g(x) ) \,dx } = \displaystyle{ \int f(x) \,dx } \pm \displaystyle{ \int g(x) \,dx }$ .

Because the integral

$\displaystyle{ \int e^{kx} \,dx }$ ,

where k is any nonzero constant, appears so often in the following set of problems, we will find a formula for it now using u-substitution so that we don't have to do this simple process each time. Begin by letting

u=kx

so that

du = k dx ,

(1/k)du = dx .

Now substitute into the original problem, replacing all forms of x, and getting

$\displaystyle{ \int e^{kx} \,dx } = \displaystyle{ \int e^{u} \, (1/k) du }$

$= \displaystyle{ (1/k) \int e^{u} \, du }$

$= \displaystyle{ (1/k) e^{u} + C }$

$= \displaystyle{ (1/k) e^{kx} + C }$ .

We now have the following variation of formula 1.) :

3. $\displaystyle{ \int e^{kx} \,dx } = \displaystyle{ (1/k) e^{kx} + C }$ .

The following often-forgotten, misused, and unpopular rules for exponents will also be helpful :

$\displaystyle{ R^M R^N = R^{M+N}}$

and

$\displaystyle{ { R^M \over R^N } = R^{M-N}}$ .

Most of the following problems are average. A few are challenging. Knowledge of the method of u-substitution will be required on many of the problems. Make precise use of the differential notation dx and du and always be careful when arithmetically and algebraically simplifying expressions.

PROBLEM 1 : Integrate $\displaystyle{ \int { 5 e^x} \,dx }$ .
Click HERE to see a detailed solution to problem 1.
PROBLEM 2 : Integrate $\displaystyle{ \int ( 2-3e^x ) \,dx }$ .
Click HERE to see a detailed solution to problem 2.
PROBLEM 3 : Integrate $\displaystyle{ \int_{0}^{ (1/7) \ln 2 } 14e^{7x} \,dx }$ .
Click HERE to see a detailed solution to problem 3.
PROBLEM 4 : Integrate $\displaystyle{ \int 7^{2x+3} \,dx }$ .
Click HERE to see a detailed solution to problem 4.
PROBLEM 5 : Integrate $\displaystyle{ \int e^{5x} \Big( { e^{2x} \over 7 } + { 3 \over e^{3x} } \Big) \,dx }$ .
Click HERE to see a detailed solution to problem 5.
PROBLEM 6 : Integrate $\displaystyle{ \int { e^x (1+2e^x)^4 } \,dx }$ .
Click HERE to see a detailed solution to problem 6.
PROBLEM 7 : Integrate $\displaystyle{ \int ( e^{4x} - e^{-4x} )^2 \,dx }$ .
Click HERE to see a detailed solution to problem 7.
PROBLEM 8 : Integrate $\displaystyle{ \int e^x(1-e^x)(1+e^x)^{10} \,dx }$ .
Click HERE to see a detailed solution to problem 8.
PROBLEM 9 : Integrate $\displaystyle{ \int_{0}^{1} { 3^x+4^x \over 5^x } \,dx }$ .
Click HERE to see a detailed solution to problem 9.
PROBLEM 10 : Integrate $\displaystyle{ \int 30e^{-3x}(1+3e^{-x})^5 \,dx }$ .
Click HERE to see a detailed solution to problem 10.
PROBLEM 11 : Integrate $\displaystyle{ \int { 8e^x(3+e^x) \over \sqrt{ e^{2x}+6e^x+ 1 } } \,dx }$ .
Click HERE to see a detailed solution to problem 11.
PROBLEM 12 : Integrate $\displaystyle{ \int \big( 27e^{9x}+e^{12x} \big)^{1/3} \,dx }$ .
Click HERE to see a detailed solution to problem 12.

Click HERE to return to the original list of various types of calculus problems.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

kouba@math.ucdavis.edu

About this document ...

Duane Kouba
1999-05-15