Cambridge Additional Mathematics

cis an arbitrary constant called

the or

.

constant of integration

integrating constant

Integration (Chapter 15) 425

These rules or combinations of them can be used to differentiate all of the functions we consider in this

course. Given an algebraic formula, we can repeatedly apply these rules until we get to basic functions such

asxnorsinx, which we know how to differentiate.

However, the task of findingantiderivativesis not so easy. Given an algebraic formula there is no simple

list of rules to find the antiderivative.

The problem was finally solved in 1968 by Robert Henry Risch. He devised a method for deciding if a

function has an elementary antiderivative, and if it does, finding it. The original summary of his method

took over 100 pages. Later developments from this are now used in all computer algebra systems.

Fortunately, our course is restricted to a few special cases.

RULES FOR INTEGRATION

Forka constant,

d

dx

(kx+c)=k )

Z

kdx=kx+c

If n 6 =¡ 1 ,

d

dx

μ

xn+1

n+1

+c

¶

=

(n+1)xn

n+1

=xn )

Z

xndx=

xn+1

n+1

+c, n 6 =¡ 1

d

dx

(ex+c)=ex )

Z

exdx=ex+c

d

dx

(sinx+c) = cosx )

Z

cosxdx= sinx+c

d

dx

(¡cosx+c) = sinx )

Z

sinxdx=¡cosx+c

Function Integral

k, a constant kx+c

xn, n 6 =¡ 1

xn+1

n+1

+c

ex ex+c

cosx sinx+c

sinx ¡cosx+c

Remember that you can always check your integration by differentiating the resulting function.

4037 Cambridge

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