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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 :

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 ,

or

(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.


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



Duane Kouba
1999-05-15