When an integral contains a quadratic expression
, you can sometimes
simplify the integrand by completing the square. This eliminates the middle term of
the quadratic; the resulting integral can then
Partial fractions is the opposite of adding fractions over a common denominator. It
applies to integrals of the form
The idea is to break
(A function of the form
a rational function.)
into a sum of sm
Trig substitution reduces certain integrals to integrals of trig functions. The idea is to
match the given integral against one of the following trig identities:
If the integral contains an expression
How do you find the area of a region bounded by two curves? I'll consider two cases.
Suppose the region is bounded above and below by the two curves
and on the sides by
and
and
,
.
I divide the region
If u and v are functions of x, the Product Rule says that
Integrate both sides:
This is the integration by parts formula. The integral on the left corresponds to the
integral you're trying to do. Part
The work required to raise a weight of P pounds a distance of y feet is
footpounds. (In m-k-s units, one would say that a force of k newtons exerted over a
distance of y feet does
newton-meters, or jo
Start with an area - a planar region - which you can imagine as a piece of
cardboard. The cardboard is attached by one edge to a stick (the axis of revolution).
As you spin the stick, the area revolve
L'Hopital's Rule is a method for computing a limit of the form
c can be a number,
, or
. The conditions for applying it are:
1.
The functions f and g are differentiable in an open interval containing
For trig integrals involving powers of sines and cosines, there are two important cases:
1.
The integral contains an odd power of sine or cosine.
2.
The integral contains only even powers of sines and
Roughly speaking, an integral
is improper if:
1.
One of the limits is infinite.
2.
The integrand "blows up" somewhere on the interval of integration.
For example,
are improper because they have infini