CK-12-Calculus

http://www.ck12.org Chapter 6. Transcendental Functions

f′(x) =√^1 πσ(− 2 αk(x−x 0 ))e−αk(x−x^0 )^2

=−^2 αk√(xπσ−x^0 )e−αk(x−x^0 )^2.

Integrals Involving Exponential Functions

Associated with the exponential derivatives in the box above are the two corresponding integration formulas:

∫

budu=ln^1 bbu+C,

∫

eudu=eu+C.

The following examples illustrate how they can be used.
Example 12:
Evaluate∫ 5 xdx.
Solution:

∫

5 xdx=ln 5^15 x+C

=^5

x

ln 5+C.

Example 13:

∫

exdx.

Solution:

∫

exdx=ex+C.

In the next chapter, we will learn how to integrate more complicated integrals, such as∫x^2 ex^3 dx, with the use of
u−substitution and integration by parts along with the logarithmic and exponential integration formulas.

Multimedia Links

For a video presentation of the derivatives of exponential and logarithmic functions(4.4), see Math Video Tutorials
by James Sousa, The Derivatives of Exponential and Logarithmic Functions (8:26).