Integration of Exponentials and Logarithms

Integrating Exponentials

We have seen that the derivative of the exponential function is quite nice

d/dx ex  =  ex

Finding the integral of the exponential function is just as simple

 The Integral of the Exponential Function

Example

Integrate

x + ex dx

Solution

We integrate each term to get

1/2 x2 + ex + C

Example

Integrate

e3x + 4 dx

Solution

We use substitution here

Let

u  =  3x + 4        du  =  3 dx

We have

e3x + 4 dx     =    1/3 3e3x + 4 dx

1/3 eu du  =  1/3 eu + C  =  1/3 e3x + 4 + C

Integrals that Produce Logarithms

Earlier, we had the derivative rule

d                   1
(ln x)  =
dx                  x

We have the corresponding integration formula is

 The Integral of 1/x

Remark:  The absolute value occurs to allow x to be negative.  Since 1/x is defined for negative values of x, its integral should be also.

Example

Integrate

ex + 3/x dx

Solution

We can integrate each term to get

ex + 3 ln|x| + C

Example

Integrate

Solution

We use substitution Let

u  =  1 - ex        du  =  -ex dx

We have

Now we can integrate to get

- ln|u| + C  =  -ln|1 - ex| + C

Example

Integrate

Solution

Substitution will not work here.  Instead we use some algebra.  In algebra, we learn that we are not allowed to break apart a denominator, but we are allowed to break apart a numerator.  We have

Now we can integrate each term

=  2 ln|x| + x -1 + C

Application

The total number of fossils per cubic meter from a species of beetle that can be found in a fossil rich area of the dessert can be found by solving the differential equation

dP            50,000
=                            P(0)  =  10
dt             50 + t

and then plugging in t  =  50.   Determine how many fossils are in the area.

Solution

Solving this differential equation is equivalent to finding the antiderivative of the left hand side.  We use substitution.   Let

u  =  50 + t        du  =  dt

This gives us

50,000
du  =  50,000 ln|u| + C  =  50,000 ln|50 + t| + C
u

To find C, we use the initial condition

10  =  50,000 ln|50 + 0| + C

C  =  10 - 50,000 ln|50|  =  -195591

To find the number of fossils, plug in 50 to get

P(2)  =  50,000 ln|50 +50| - 195,591  =  10.15

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