Integration of irrational functions. Rational substitution is the usual way to integrate them. Ex. Evaluate Sol. Let then . Example. Ex. Evaluate Sol. Strategy for integration. First of all, remember basic integration formulae.
a. rational functions: partial fractions
b. rational trigonometric functions:
c. product of two different kind of functions: integration
d. irrational functions: trigonometric substitution, rational
substitution, reciprocal substitution
methods, relate the problem to known problems
function f has an antiderivative.
functions. This means, there exist functions whose
integration can not be written in terms of essential functions.
way is to find its approximate value.
left endpoint approximation
right endpoint approximation
finite interval [a,b] and the integrand f does not have an
curve from 0 to 1, we need to study the integrability
of the function on the interval [0,1].
under the curve from 1 to we need to evaluate
case where the interval is infinite and also to the case where
the integrand f has an infinite discontinuity in the interval. In
either case, the definite integral is called improper integral.
b>a, f is integrable on [a,b], then
is called the improper integral of type I of f on and
denoted by If the right side limit
exists, we say the improper integral converges.
and its convergence.
only when both and are convergent,
the improper integral converges.
the New-Leibnitz formula for improper integrals is also true:
integration by parts, are applicable to improper integrals.
Especially, if an improper integral can be converted into a
proper integral by substitution, then the improper integral
continuous on [a,b) and x=b is a vertical asymptote ( b is
said to be a singular point ), then
is called the improper integral of type II. If the limit exists,
we say the improper integral converges.
Only when both of the two improper integrals and
converge, the improper integral converge.
integration by parts are all true for improper integrals of
continuous functions with for then
(a)If converges, then converges.
(b)If the latter diverges, then the former diverges.
hold true for improper integrals.
Ex. Thefunction defined by the improper integral
is called Gamma function. Evaluate