Integration of Exponentials and Logarithms

Integrating Exponentials

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

        d/dx e^x  =  e^x

Finding the integral of the exponential function is just as simple

Example

Integrate

        x + e^x dx

Solution

We integrate each term to get

        1/2 x² + e^x + C

Example

Integrate

        e^{3x + 4} dx

Solution

We use substitution here

        Let  

        u  =  3x + 4        du  =  3 dx

We have

        e^{3x + 4} dx     =    1/3 3e^{3x + 4} dx

        1/3 e^u du  =  1/3 e^u + C  =  1/3 e^{3x + 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

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

        e^x + 3/x dx

Solution

We can integrate each term to get

        e^x + 3 ln|x| + C

Example

Integrate

Solution

We use substitution.  Let  

        u  =  1 - e^x        du  =  -e^x dx

We have

Now we can integrate to get

        - ln|u| + C  =  -ln|1 - e^x| + 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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