2/3
categories of problems:
trig substitution
other substitution
"Let u = anything weird"
partial fractions
might have to do completion of the square first
might have to do long division first
integration by parts
look for
Int f(x)g(x) dx
where
one of tho
1/13
overview:
integration
techniques of antidifferentiation
example:
applications of integrals
example:
particularly, power series
example:
example:
e^x = Sum_cfw_n = 0 to infinity (x^n)/n!
differential equations
volume of a body of revolution
sequences
1/14
hints for using substitution
When looking at
remember that you want to express
Int g(x) dx
g(x) = f(u) du/dx
for some function u
That means you must look to express the integrand (i.e., g(x) as a product
one factor is something you can write as the d
1/15
some "interesting" uses of integration by parts:
Int ln(x) dx =
u = ln(x),
du = 1/x dx,
= x ln(x) - Int dx
= (ln(x)x - Int x (1/x) dx
= x ln(x) - x
In general, the same method works for
Int poly(x) ln(x) dx
for any polynomial
poly(x)
by using
v=x
= u
1/22
Partial Fractions:
make sure your fraction, Top/Bottom
is proper (using long division, if necessary)
has Bottom factored into powers of
linear factors
quadratic factors
express Top/Bottom in terms of the proper framework:
sum of fractions, each one o
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Partial Fractions Frameworks revisited:
For every factor raised to a power in the denominator of the target:
make sure that factor and that power is in the denominator of one of the
partial fractions
So if you're trying to simplify
(x^2 + 1)/(x^3(x^2
trig substitutions revisited
determine the basic pattern:
1 - x^2:
x^2 -1
secant substitution
1 + x^2
sine substitution
tangent substitution
basic pattern is determined by + and -, not by specific coefficients
factor out the constant to get a 1
2 - 3x^2 =
1/17
using tables of integrals:
reduction formulas
These express an integral in terms of another, simpler one.
Most of them are very straight-forward; the only one requiring care is the
one for
sin^m(x) cos^n(x)
m or n odd:
Int sin^5(x) cos^2(x) dx =
= -
1/21
care with completing the square
Note that you must be careful when factoring out a constant to make the x^2 term
have coefficient 1:
Int 1/(2x^2 + x)^1/2 dx = Int 1/(2(x^2 + 1/2)^1/2 dx
= (1/2^1/2) Int du/(u^2 - 1/16)^1/2
(u = x+ 1/4, du = dx)
= Int