Mathematics by T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel (chapter 11 / 12: Integral calculus)restart; Integral calculusIn Maple the anti-derivative of an expression can be determinded by the command int .int(x*ln(x),x);If a range is set for the integration variable, the integral will be evaluated. int(ln(x)/sqrt(x),x=0..1);Obviously unbounded functions and/or intervals are allowed. We can also determine the Laplace transformation of 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 . assume(s>0);
int(t*exp(-s*t),t=0..infinity);Restrictions for s are required, to make sure that the integral exists.A lot of integrals cannot be evaluated in a closed representation. Thus we obtain int(sin(x)/x,x);
int(exp(-x^2),x);(erf,). If we set interval boundaries in such cases, the according expression is returned. int(sin(x)/sqrt(x),x=0..1);We reach a numerical evaluation byevalf(int(sin(x)/sqrt(x),x=0..1));(see int[numerical]). A numerical evaluation of an integral may be forced without any previous algebraic examination. Therefore the command Int, that first does not provide an evaluation. In combination with evalf a numerical approximation method is applied.Int(sqrt(x)/sin(x),x=0..1);
evalf(%);Usually the Clenshaw-Curtis method is applied if no specifications are done. Other methods (e.g. a Newton-Cotes formula) are optional.Furthermore the accuracy can be set by the number of decimal places (here 20).evalf(Int((x^3+1)/(x^5-x^2+1),x=0..1, 20, _NCrule));In the package "student" the trapezoidial and the Simpson rule can be found. with(student);(Remark: Unfortunately the command Int is slightly modified by now, so that the numerical options from above are no longer accepted. You can undo this by restart .).trapezoid(x^n*sin(x),x=0..Pi);
simpson(x^n*sin(x),x=0..Pi);The amount of base points are set by an additional argument.simpson(x^3*sin(x),x=0..Pi,20);
evalf(%);Using a loop, we can easily generate a table and study the convergency behaviour,a:=evalf(1/4*(1-exp(-4))): for j from 2 to 6 do
2^j, abs(evalf(trapezoid(exp(-4*x),x=0..1,2^j))-a), abs(evalf(simpson(exp(-4*x),x=0..1,2^j))-a);
od;Regarding the partial fraction decomposition method, it is useful to revisit transformations of rational expressions. Let us consider the following exemplary expressionr:=(x^3+2*x^2-x+1)/(x^3-2*x^2-x+2);We obtain the partial fraction decomposition of r by convert including the according option.convert(r,parfrac,x);To check each single step, we first define the numerator and denominator polynominal by numer and denom .p:=numer(r); q:=denom(r);A long division is given byr1:=quo(p,q,x) + rem(p,q,x)/q;while quo evaluates the quotient and rem the remainder. We verify the result withevalb(r=normal(r1)); Beside the command normal other transformation commands are useful, e.g.expand(r);(see further advices in the help pages). The numerator is factorized byfactor(q); Exercises(To work on the exercises, please open a new worksheet to trial an error your commands. You will get a suggestion for the solution, if you open up the solution. Before you look at them try on your own.)1. Evaluate the integrals
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, 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, 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Solutionint(x*ln(x+3),x=0..1);int((ln(x))^2,x=1..2);int(1/(sqrt(x+2)*(3-x)),x=-1..1); evalf(%);2. Determine the anti-derivatives for the following functions, a) f(x) = 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 , b) f(x) = 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 ,
c) f(x) = 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 , d) f(x) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1GIzYoLUkjbWlHRiQ2I1EhRictRiM2KC1JJW1zdXBHRiQ2JS1JI21vR0YkNi1RLyZFeHBvbmVudGlhbEU7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjExMTExMTFlbUYnLUYjNiotSSNtbkdGJDYkUSIyRidGPS1GOjYtUTEmSW52aXNpYmxlVGltZXM7RidGPUZARkNGRUZHRklGS0ZNRk8vRlNGUS1GMTYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvRj5RJ2l0YWxpY0YnRmVuLUYxNiVRInhGJ0Zcb0Zfby8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GOjYtUSIrRidGPUZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRmFwLUZYNiRRIjFGJ0Y9RmRvRmdvRj1GMEZkb0Znb0Y9LUYjNihGMC1GIzYoLUY3NiVGOS1GIzYoRmluRmVuRmFvRmRvRmdvRj1Gam9GXXBGY3BGZG9GZ29GPUYwRmRvRmdvRj0vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNxLyUpYmV2ZWxsZWRHRkJGZG9GZ29GPQ== . Solutionint((5*x^2-11*x+5)/(x^3-4*x^2+5*x-2),x);int((x^3+6*x^2+3*x+18)/(x^3+x^2+4*x+4),x);int((cos(x)+sin(x)+1)/(1+cos(x))/(1-cot(x/2)),x);a:='a':
int((exp(2*a*x)+1)/(exp(a*x)+1),x);3. Make a long devision LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1GIzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GOFEnbm9ybWFsRictRiM2Ji1GMTYlUSJxRidGNEY3RjpGPUZALyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZMLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjpGPUZA of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictRiM2OUYrLUYjNjctRiw2NVEicEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUqdW5kZXJsaW5lR0Y+LyUqc3Vic2NyaXB0R0Y+LyUsc3VwZXJzY3JpcHRHRj4vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGPi8lK2V4ZWN1dGFibGVHRj4vJSlyZWFkb25seUdGPi8lKWNvbXBvc2VkR0Y+LyUqY29udmVydGVkR0Y+LyUraW1zZWxlY3RlZEdGPi8lLHBsYWNlaG9sZGVyR0Y+LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y+LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0ZpblEnbm9ybWFsRicvJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZhcC1JKG1mZW5jZWRHRiQ2NS1GIzY1LUYsNjVRInhGJ0Y2RjlGPEY/RkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRjZGOUY8L0ZARj5GQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX28tRlxvNj5RIj1GJ0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0Zhb0Zjb0Zlb0Znb0Zpb0ZbcEZdcC9GYHBRLDAuMjc3Nzc3OGVtRicvRmNwRmFxLUYjNjxGKy1GIzY1LUklbXN1cEdGJDYlRmlwLUkjbW5HRiQ2NVEiNUYnRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfby1GXG82PlEiK0YnRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vRmFvRmNvRmVvRmdvRmlvRltwRl1wL0ZgcFEsMC4yMjIyMjIyZW1GJy9GY3BGZXItRiM2NS1GaHE2JUZpcC1GW3I2NVEiM0YnRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vRl5yRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vLUZcbzY+USgmbWludXM7RidGNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GYW9GY29GZW9GZ29GaW9GW3BGXXBGZHJGZnItRiM2Ny1GW3I2NVEiMkYnRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vLUZcbzYtUTEmSW52aXNpYmxlVGltZXM7RidGX29GYW9GY29GZW9GZ29GaW9GW3BGXXBGX3BGYnBGaXBGNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GYXItRltyNjVRIjFGJ0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0YrRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vRitGNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX28= und LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictRiM2OUYrLUYjNjctRiw2NVEicUYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUqdW5kZXJsaW5lR0Y+LyUqc3Vic2NyaXB0R0Y+LyUsc3VwZXJzY3JpcHRHRj4vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGPi8lK2V4ZWN1dGFibGVHRj4vJSlyZWFkb25seUdGPi8lKWNvbXBvc2VkR0Y+LyUqY29udmVydGVkR0Y+LyUraW1zZWxlY3RlZEdGPi8lLHBsYWNlaG9sZGVyR0Y+LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y+LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0ZpblEnbm9ybWFsRicvJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZhcC1JKG1mZW5jZWRHRiQ2NS1GIzY1LUYsNjVRInhGJ0Y2RjlGPEY/RkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRjZGOUY8L0ZARj5GQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX28tRlxvNj5RIj1GJ0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0Zhb0Zjb0Zlb0Znb0Zpb0ZbcEZdcC9GYHBRLDAuMjc3Nzc3OGVtRicvRmNwRmFxLUYjNjxGKy1GIzY1LUklbXN1cEdGJDYlRmlwLUkjbW5HRiQ2NVEiM0YnRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfby1GXG82PlEiK0YnRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vRmFvRmNvRmVvRmdvRmlvRltwRl1wL0ZgcFEsMC4yMjIyMjIyZW1GJy9GY3BGZXItRiM2NS1GaHE2JUZpcC1GW3I2NVEiMkYnRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vRl5yRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vLUZcbzY+USgmbWludXM7RidGNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX29GYW9GY29GZW9GZ29GaW9GW3BGXXBGZHJGZnJGaXBGXnMtRltyNjVRIjFGJ0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0Y2RjlGPEZccUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZfb0YrRjZGOUY8RlxxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRl9vRitGNkY5RjxGXHFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GX28= and determine the partial fraction decomposition for p/q.SolutionAfter entering the polynominalsp:=x^5+x^3-2*x+1;
q:=x^3+x^2-x-1;we determinequo(p,q,x);rem(p,q,x);and obtain the partial fraction decomposition byconvert(p/q,parfrac,x);