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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 53 "Some Basic Formulas for \+
computing the antiderivatives" }}{PARA 0 "" 0 "" {TEXT -1 333 "Maple V
 is an excellent tool for computing antiderivatives.  In this section \+
 Maple V will be used to establish some basic formulas. Remember the p
roblem in finding an antiderivative for  f(x) is to find a function F(
x) such that F'(x)=f(x).  Thus one can always determine the correctnes
s of the  antiderivative by differentiation.  " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "Int(x^n,x): % = value(%) + C;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 95 "illustrates that if  n  is not -1 then the general a
ntiderivative or the indefinite integral of" }}{PARA 0 "" 0 "" {TEXT 
-1 65 "x^n is  x^(n+1)/(n+1)+C.  This can be checked by differentiatio
n." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(rhs(%),x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 72 "What happens in the case that n=-1?   What is the an
tiderivative of 1/x?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(subs(n=
-1,x^n),x): % = value(%)+C;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We
 check this result by differentiation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "diff(rhs(%),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "di
ff(subs(x=-x,ln(x)),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "For th
e sin and cos functions we have" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "
Int(sin(x),x): %=value(%)+C;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "Int(cos(x),x):%=value(%) +C;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 46 "Example 2:   Evaluate the following integrals:" }}{PARA 0 "" 0 
"" {TEXT -1 66 "int \\cos(sqrt(x)), x),   int(cos(x^2) ,x),  int(cos(x
^2), x=0..1)." }}{PARA 0 "" 0 "" {TEXT -1 9 "Solution:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "Int(cos(sqrt(x)),x) : %=value(%)+C;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(rhs(%),x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Int(cos(x^2),x):% = value(%)+C;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "You will not be expected to know \+
the properties of the function called FresnelC in this course." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(rhs(%),x);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 34 "Int(cos(x^2),x=0..1): %= value(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(cos(x^2),x=0..1) = evalf
(rhs(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 \+
1 0" 333 }{VIEWOPTS 1 1 0 1 1 1803 }