integration exponential

∫ 4x2ex^2dx
=∫ 2xex^2dx2
=∫ 2xd(ex^2) (since dey=eydy)
=2x ex^2-∫ 2ex^2dx
=2x ex^2- 2Erf(x)

There is no analytic form of the integral of ∫ex^2dx, this integral is defined as Erf(x). Integrate ∫ex^2dx over x=-infinity to infinity =(2π)1/2

The integral of e^(x^2) is not exactly Erf(x), see
http://mathworld.wolfram.com/Erf.html

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 :
,


where , and

,


where a is any positive constant not equal to 1 and 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 :

,


where n is any constant not equal to -1,

,


where k is any constant, and

.


Because the integral

,


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







.


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

3. .


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




and

.


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 .
Click HERE to see a detailed solution to problem 1.




PROBLEM 2 : Integrate .
Click HERE to see a detailed solution to problem 2.




PROBLEM 3 : Integrate .
Click HERE to see a detailed solution to problem 3.




PROBLEM 4 : Integrate .
Click HERE to see a detailed solution to problem 4.




PROBLEM 5 : Integrate .
Click HERE to see a detailed solution to problem 5.




PROBLEM 6 : Integrate .
Click HERE to see a detailed solution to problem 6.




PROBLEM 7 : Integrate .
Click HERE to see a detailed solution to problem 7.




PROBLEM 8 : Integrate .
Click HERE to see a detailed solution to problem 8.




PROBLEM 9 : Integrate .
Click HERE to see a detailed solution to problem 9.




PROBLEM 10 : Integrate .
Click HERE to see a detailed solution to problem 10.




PROBLEM 11 : Integrate .
Click HERE to see a detailed solution to problem 11.




PROBLEM 12 : Integrate .
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 :

[email protected]






--------------------------------------------------------------------------------

About this document ...
Duane Kouba
1999-05-15