{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helveti ca" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {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 }