The Handy Math Answer Book

INTEGRAL CALCULUS

What is the integral calculus?

The integral calculus is the part of “the” calculus that deals with integrals—both the inte-

gral as the limit of a sum and the integral as the antiderivative of a function (see below for

more information). In general, the integral calculus is the limit of a sum of elements in

which the number of the elements increase without bound, while the size of the elements

diminishes. It is also considered the second most important kind of limit in the calculus

(the first being limits in association with derivatives). It was originally developed by using

polygons to approximate areas of geometrically shaped objects such as circles.

What are some common integralsin the calculus?

There are many standard integrals used in the calculus. The following lists only a few

of those integrals in their common form:

() ()

()

{()} Fudu

dx

du fx

Fu

###Ffx dx==l du

##faxdx()=a^1 fudu()

udv=-uv ## vdu

#()u v w!! !gdx= udx!! !vdx ### wdx g

af x dx() =a## f x dx()

#adx=ax

dxd (())cscu ddx

uu

u

1

1 1

=- (^2) -

dxd (())secu ddx

uu

u

1

1 1

= (^2) -

dxd (())cot-^1 u = 1 -^1 udx^2 du

(())tan

dx

d u

udx

du

1

1 1

• = - 2

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