We often integrate over an infinite range. Such integrals are called improper. They are defined a s limits,
Example 5.7:
Evaluate .
Solution:
Apply the definition
A inite integral is called convergent, if tyhe limit does not exist the integral is called divergent.
Let us look at a physics example
Example 5.8:
Determine the escape velocity from earth.
Solution:
We need belance of energies. The initial kinetic energy must equal the work done against gravity to get the object )of mass to escape from the gravity field of the earth (mass , radius )
Evaluate the integral as above,
Integrals that require special attentions is those where the integrand diverges. We need to take a start-point just above and below the singularity, and take a limit. A simple and obvious example is
Be extrememly careful when the singularity occurs in the middle of the integration interval.
Example 5.9:
Calculate
Solution:
Split the interal into two parts,
The naive answer is 2! So we note that we have to be extremely careful