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Integral calculus
  Definite integrals
 Substitution and definite integration
When we evaluate an indefinite integral of the form
we use the substitutions  g (x) = u  and  g' (x) dx = du  to obtain
Therefore, to evaluate the definite integral using substitution
or
Note, by making substitution the definite integral changes the limits of integration.
That is, the new limits of integrations must correspond to the range of values for u as we now integrate the simpler integrand   f (u)  from  ua = g (a)  to  ub = g (b).
 Substitution and definite integration examples
 Example:  Evaluate
 Solution:
 
 
 
 Example:  Evaluate
 Solution:
 Integrations by parts and the definite integral
If u (x) and v (x) are continuous functions with continuous derivatives on [a, b], then
 Example:  Evaluate
 Solution:
  The improper integrals
A definite integral with one or both limits of integration infinite, or having an integrand that becomes infinite between the limits of integration is called the improper integral.
Let  f (x) be defined on [a, oo ) and integrable on [a, b] for all  a < b < oo, then
provided the limit exists.
If  f (x) is defined on ( - oo, b] and integrable on [a, b] for all  - oo < a < b, then
provided the limit exists.
If  f (x) is defined on ( - oo, oo ) and integrable on any closed interval [a, b], then
provided the limits on the right exist.
In each case where the appropriate limit exists we say the integral converges, otherwise the integral is said to diverge.
 Example:  Evaluate
 Solution:
  As the area of the region beneath the graph of  f over the interval [1, b] as b goes to infinity A = 1, thus the integral converges.
 Example:  Evaluate
 Solution:
Therefore, the integral diverges.
 Example:  Evaluate
 Solution:
Therefore, the integral converges.
 Example:  Evaluate
 Solution:
Therefore, the integral converges.
 Example:  Evaluate
 Solution:
 
Therefore, the integral converges.
 
 
 
 
 
 
 
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