Lesson 18: Elementary antiderivatives for logarithmic functions restart;

The procedure for integrating rational functions is guaranteed to always work: it always produces an elementary antiderivative for the rational function. Now I want to look at one case where there may or may not be an elementary antiderivative: a function that is rational except for one logarithmic term. The logarithmic term will be 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRjZGOUZARkBGQEYrRkBGK0ZA is a rational function. I'll suppose our integrand is 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEnJiM5NTI7RicvRjdGREZARkBGQEZARitGQEYrRkA= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEnJiM5NTI7RicvRjdGREZARkBGQEZARitGQEYrRkA= are polynomials with coefficients that can be rational functions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo=, with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEnJiM5NTI7RicvRjdGREZARkBGQEZARitGQEYrRkA= of lower degree than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEnJiM5NTI7RicvRjdGREZARkBGQEZARitGQEYrRkA=, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEnJiM5NTI7RicvRjdGREZARkBGQEZARitGQEYrRkA= is square-free, i.e. it can't be divided by the square of any non-constant polynomial. Here are some examples: 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 with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y5LyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1GIzYmLUYsNiVRI2xuRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EmMC4wZW1GJy9GUkZmbi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GN1EldHJ1ZUYnL0Y6USdpdGFsaWNGJ0Y5RjlGOUYrRjlGK0Y5RitGOQ==, 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, 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. This does have an elementary antiderivative, and a Math 101 student might be able to find it. int(1/(x*ln(x)), x);

LUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1GIzYjSSJ4R0Yn

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 with 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, 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, 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. This does not have an elementary antiderivative. int(x/ln(x), x);

LCQtSSNFaUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkIiIiLCQtSSNsbkdGJTYjSSJ4R0YoISIjISIi

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 with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y5LyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1GIzYmLUYsNiVRI2xuRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EmMC4wZW1GJy9GUkZmbi1JKG1mZW5jZWRHRiQ2JC1GIzYmRistRiM2J0YrLUYjNiQtSSVtc3VwR0YkNiUtRiw2JVEieEYnL0Y3USV0cnVlRicvRjpRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjktRj02LVEiK0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EsMC4yMjIyMjIyZW1GJy9GUkZmcC1GXHA2JFEiMUYnRjlGOUYrRjlGOUY5RitGOUYrRjlGK0Y5, 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, 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. This one is not so obvious, but int(((x^2+1)*ln(x^2+1) - 2*x^2)/((x^2+1)*(ln(x^2+1)^2 - x^2)), x);

LCYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCZJInhHRigiIiItRiQ2IywmKiRGKyIiI0YsRixGLCEiIiNGMkYxLUYkNiMsJkYrRixGLUYsI0YsRjE=

The method is somewhat analogous to what we did for the "logarithmic part" of rational functions: 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRjZGOUZARkBGQEYrRkBGK0ZAwas square-free and had higher degree than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRjZGOUZARkBGQEYrRkBGK0ZA. There we looked at the resultant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRjZGOUZARkBGQEYrRkBGK0ZA and 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 with respect to x, which is a polynomial in z with constant coefficients, and we found an antiderivative that was the sum of 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 for all roots LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImpGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== of that resultant, where 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. Here, the Rothstein-Trager Theorem says Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiekYnRjZGOUZARkBGQEYrRkBGK0ZA be the resultant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEnJiM5NTI7RicvRjdGREZARkBGQEZARitGQEYrRkA= and 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 with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3. Note that this is a polynomial in z, whose coefficients can be rational functions in x. 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 has an elementary antiderivative if and only if all the roots of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiekYnRjZGOUZARkBGQEYrRkBGK0ZA are constants (i.e. don't depend on x). If they are constants, let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRkMvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZDRitGQw==, ..., LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtRiw2JVEibkYnRjdGOi9GO1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZCRitGQg== be those roots, and 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 (as polynomials in \316\270). Then 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. One more point. In the gcd command, we don't say what the variable is, even if there are two possibilities (x and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3): gcd actually works for polynomials in several variables. But sometimes we might have polynomials in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3 with coefficients that are rational functions (but not polynomials) in x. Then gcd would complain: gcd(theta/x, theta^2 + theta);

Error, (in gcd/Freeze) arguments should be polynomials

The cure for this is to use gcdex instead. The ex stands for extended. You give gcdex the variable name as well as the two polynomials: gcdex(theta/x, theta^2 + theta, theta);

SSZ0aGV0YUc2Ig==

Let's see how this works in each of the examples I mentioned. Example 1: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJm1mcmFjR0YkNigtRiM2JC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOi1GIzYmRistRiM2Jy1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjtRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjovJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk8vJSlzdHJldGNoeUdGTy8lKnN5bW1ldHJpY0dGTy8lKGxhcmdlb3BHRk8vJS5tb3ZhYmxlbGltaXRzR0ZPLyUnYWNjZW50R0ZPLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGaG4tRiM2Ji1GLDYlUSNsbkYnL0ZFRk9GOi1GSjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y6Rk1GUEZSRlRGVkZYRlpGZm5GaW4tSShtZmVuY2VkR0YkNiQtRiM2JEZBRjpGOkY6RitGOkYrRjovJS5saW5ldGhpY2tuZXNzR0Y5LyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl1wLyUpYmV2ZWxsZWRHRk9GOkYrRjo= with 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, 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, 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 A:= 1; B:= x*ln(x); C:= A - z*diff(B, x);

IiIi

KiZJInhHNiIiIiItSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiNGJQ==

LCYiIiJGIyomSSJ6RzYiRiMsJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNJInhHRiZGI0YjRiNGIyEiIg==

R:=resultant(x*theta, 1-z*(theta+1), theta);

KiZJInhHNiIiIiIsJkYlRiVJInpHRiQhIiJGJQ==

solve(R,z);

IiIi

That's constant, so yes, this has an elementary antiderivative. There's only one G. G:= gcdex(x*theta, 1-1*(theta+1),theta);

SSZ0aGV0YUc2Ig==

And the antiderivative is: F := eval(1*ln(G),theta=ln(x));

LUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1GIzYjSSJ4R0Yn

Example 2: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJm1mcmFjR0YkNigtRiM2JC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y9USdub3JtYWxGJy1GIzYmRistRiM2Ji1GLDYlUSNsbkYnL0Y6USZmYWxzZUYnRj8tSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y/LyUmZmVuY2VHRkkvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGaG4tSShtZmVuY2VkR0YkNiRGNEY/Rj9GK0Y/LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zjby8lKWJldmVsbGVkR0ZJRj9GK0Y/ with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y5LyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1GIzYmLUYsNiVRI2xuRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EmMC4wZW1GJy9GUkZmbi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GN1EldHJ1ZUYnL0Y6USdpdGFsaWNGJ0Y5RjlGOUYrRjlGK0Y5RitGOQ==, 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, 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. A:= x; B:= ln(x); C:= A - z*diff(B, x);

SSJ4RzYi

LUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGJw==

LCZJInhHNiIiIiIqJkkiekdGJEYlRiMhIiJGKA==

R:=resultant(theta, x - z/x, theta);

KiYsJiokSSJ4RzYiIiIjIiIiSSJ6R0YmISIiRihGJUYq

solve(R,z);

KiRJInhHNiIiIiM=

That's not constant, so the answer is no: this one doesn't have an elementary antiderivative. Example 3: 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 with 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, 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, 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 A:= (x^2+1)*ln(x^2+1)-2*x^2; B:= (x^2+1)*(ln(x^2+1)^2-x^2); C:= A - z*diff(B,x);

LCYqJiwmKiRJInhHNiIiIiMiIiJGKUYpRiktSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjRiRGKUYpRiUhIiM=

KiYsJiokSSJ4RzYiIiIjIiIiRihGKEYoLCYqJC1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGI0YnRihGJCEiIkYo

LCgqJiwmKiRJInhHNiIiIiMiIiJGKUYpRiktSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjRiRGKUYpRiUhIiMqJkkiekdGJ0YpLCYqJkYmRiksJiokRipGKEYpRiUhIiJGKUYoKiZGJEYpLCYqKEYqRilGJkYpRiRGNyIiJUYmRjBGKUYpRilGNw==

Atheta:= eval(A,ln(x^2+1)=theta); Btheta:= eval(B,ln(x^2+1)=theta); Ctheta:= eval(C,ln(x^2+1)=theta);

LCYqJiwmKiRJInhHNiIiIiMiIiJGKUYpRilJJnRoZXRhR0YnRilGKUYlISIj

KiYsJiokSSJ4RzYiIiIjIiIiRihGKEYoLCYqJEkmdGhldGFHRiZGJ0YoRiQhIiJGKA==

LCgqJiwmKiRJInhHNiIiIiMiIiJGKUYpRilJJnRoZXRhR0YnRilGKUYlISIjKiZJInpHRidGKSwmKiZGJkYpLCYqJEYqRihGKUYlISIiRilGKComRiRGKSwmKihGKkYpRiZGKUYkRjIiIiVGJkYrRilGKUYpRjI=

R:=resultant(Btheta, Ctheta, theta);

KigsJiokSSJ4RzYiIiIjIiIiRihGKEYnRiVGJywuISIiRigqJkYlIiIlSSJ6R0YmRidGLComRiVGJ0YtRichIikqJEYtRidGLEYkRicqJEYlRixGKkYo

solve(R,z);

NiQjIiIiIiIjIyEiIkYl

Both are constants, so the answer is yes. G1 := gcdex(Btheta, eval(Ctheta, z=1/2),theta);

LCZJJnRoZXRhRzYiIiIiSSJ4R0YkRiU=

G2 := gcdex(Btheta, eval(Ctheta, z=-1/2),theta);

LCZJJnRoZXRhRzYiIiIiSSJ4R0YkISIi

And the antiderivative is: F:= eval(1/2*ln(G1)-1/2*ln(G2), theta=ln(x^2+1));

LCYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCZJInhHRigiIiItRiQ2IywmKiRGKyIiI0YsRixGLEYsI0YsRjEtRiQ2IywmRi1GLEYrISIiI0Y2RjE=

When you want some idea of how Maple does some computation, the infolevel facility may help. For information on a particular command f, you give a value (an integer from 1 to 5) to infolevel[f]. For example, for information on integration you could try infolevel[int] := 3;

IiIk

int(1/(1+x^2+x^5),x);

int/indef1: first-stage indefinite integration int/ratpoly: rational function integration int/ratpoly/horowitz: integrating

KiQsKCIiIkYkKiRJI19YRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiIiIjRiQqJEYmIiImRiQhIiI=

int/ratpoly/horowitz: Horowitz' method yields

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCwoIiIiRisqJEkjX1hHRiQiIiNGKyokRi0iIiZGKyEiIkYt

int/risch/ratpoly: starting computing subresultants at time 1.341 int/risch/ratpoly: end of subresultants computation at time 1.341 int/risch/ratpoly: Rothstein's method - factored resultant is

NyM3JCwqKiRJInpHNiIiIiYiIiIqJEkiekc2IiIiJCMiI0YiJUxLKiRJInpHNiIiIiMjIiNdIiVMSyMhIiIiJUxLIiIiIiIi

int/risch/ratpoly: result is

LUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJkkjX1JHRiciIiItSSNsbkdGJDYjLC5JI19YR0YkRisqJEYqIiIlIyErRSUpcENBIihmUE8oKiRGKiIiJCMiKkRgLCh6RjUqJEYqIiIjIyEqPnVdNCJGNUYqIyIpMHckKT1GNSMiKCs/NChGNUYrRisvRiotSSdSb290T2ZHRiQ2IywqKiRJI19aR0YkIiImIiVMSyokRkhGNyIjRiokRkhGOyIjXSEiIkYr

LUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJkkjX1JHRiciIiItSSNsbkdGJDYjLC5JInhHRidGKyokRioiIiUjIStFJSlwQ0EiKGZQTygqJEYqIiIkIyIqRGAsKHpGNSokRioiIiMjISo+dV00IkY1RiojIikwdyQpPUY1IyIoKz80KEY1RitGKy9GKi1JJ1Jvb3RPZkdGJDYjLCoqJEkjX1pHRiQiIiYiJUxLKiRGSEY3IiNGKiRGSEY7IiNdISIiRis=

The basic scheme is: Level 1: reserved for information that the user must be told. Level 2,3: general information, including technique or algorithm being used. Level 4,5: more detailed information about how the problem is being solved However, the usefulness of this facility is rather uneven: it's up to the Maple programmers to decide what to tell, and this varies a lot from one command to another. By the way, in many cases (such as int), if you try setting the infolevel and then repeating a command you already tried, you won't get anything from infolevel, because Maple just remembers the result it got last time rather than doing it all over again. In such a case you might want to use restart to get rid of all remembered results. restart; infolevel[int]:= 1; int(1/(1+x^2+x^5),x);

IiIi

int/indef1: first-stage indefinite integration int/ratpoly: rational function integration

LUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJkkjX1JHRiciIiItSSNsbkdGJDYjLC5JInhHRidGKyokRioiIiUjIStFJSlwQ0EiKGZQTygqJEYqIiIkIyIqRGAsKHpGNSokRioiIiMjISo+dV00IkY1RiojIikwdyQpPUY1IyIoKz80KEY1RitGKy9GKi1JJ1Jvb3RPZkdGJDYjLCoqJEkjX1pHRiQiIiYiJUxLKiRGSEY3IiNGKiRGSEY7IiNdISIiRis=

If you don't know which command you need to know about, you might try infolevel[all], which will apply to all Maple commands. infolevel[all]:= 1; int(x*ln(x)/(1+ln(x)^2), x);

IiIi

int/indef1: first-stage indefinite integration int/indef2: second-stage indefinite integration int/ln: case of integrand containing ln int/rischnorm: enter Risch-Norman integrator radfield: computing a basis for {} radfield: over {} radfield: options are {} radfield: computing a basis for {} radfield: over {} radfield: options are {} factor/polynom: polynomial factorization: number of terms 2 factor/polynom: polynomial factorization: number of terms 2 solve: Warning: no solutions found int/rischnorm: exit Risch-Norman integrator int/risch: enter Risch integration radfield: computing a basis for {} radfield: over {} radfield: options are {} radfield: computing a basis for {} radfield: over {} radfield: options are {} radfield: computing a basis for {} radfield: over {} radfield: options are {} radfield: computing a basis for {} radfield: over {} radfield: options are {} int/risch: exit Risch integration

LUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKEkieEdGJyIiIi1JI2xuR0YkNiNGKkYrLCZGK0YrKiRGLCIiI0YrISIiRio=

To turn off the use of infolevel for a particular command, you can set it to 0. infolevel[all] := 0; infolevel[int] := 0;

IiIh

IiIh

int(1/(x^11+1),x);

LCYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCZJInhHRigiIiJGLEYsI0YsIiM2LUkkc3VtR0YlNiQqJkkjX1JHRihGLC1GJDYjLCZGK0YsRjNGLEYsL0YzLUknUm9vdE9mR0YlNiMsOCokSSNfWkdGJSIjNUYsKiRGPSIiKkYsKiRGPSIiKUYsKiRGPSIiKEYsKiRGPSIiJ0YsKiRGPSIiJkYsKiRGPSIiJUYsKiRGPSIiJEYsKiRGPSIiI0YsRj1GLEYsRixGLQ==

Maple has some very good numerical methods for calculating definite integrals. As we've seen, these will work even when int does not return a value. infolevel[evalf]:= 3; evalf(Int(x/(sin(x)*(1+cos(x)^2)), x= 0..Pi/2));

IiIk

evalf/int/control: integrating on 0 .. 1/2*Pi the integrand

KihJInhHNiIiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YjISIiLCZGJUYlKiQtSSRjb3NHRihGKyIiI0YlRiw=

evalf/int/control: tolerance = .5000000000e-9; method = _DEFAULT; maxintervals = _DEFAULT Control: Entering NAGInt Control: trying d01ajc (nag_1d_quad_gen) d01ajc: epsabs=.500000000000000e-12; epsrel=.5000000000e-9; max_num_subint=200 d01ajc: result=1.33861101927154235 d01ajc: abserr=.628550758933998994e-11; num_subint=1; fun_count=21 Control: result=1.33861101927154235

JCIrPjVoUTghIio=

NAG stands for Numerical Algorithms Group, which wrote lots of the numerical algorithms that Maple uses, especially those that use "hardware" floating point, i.e. the built-in binary floating-point arithmetic that comes with the computer, rather than Maple's usual arbitrary-precision decimal "software" floating point. Hardware floating-point arithmetic is much faster, and in particular the NAG algorithms are highly optimized and very fast. In this case Maple used a NAG algorithm called d01ajc. It returned a hardware-float result of 1.33861101927154235, which Maple then rounded to 10 digits as 1.338611019. evalhf(Digits);

JCIjOiIiIQ==

That's roughly the Digits equivalent of hardware floats on this machine. If we asked for more than that, Maple would not use the NAG algorithm. evalf(Int(x/(sin(x)*(1+cos(x)^2)), x= 0..Pi/2),16);

evalf/int/control: integrating on 0 .. 1/2*Pi the integrand

KihJInhHNiIiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YjISIiLCZGJUYlKiQtSSRjb3NHRihGKyIiI0YlRiw=

evalf/int/control: tolerance = .5000000000000000e-15; method = _DEFAULT; maxintervals = _DEFAULT evalf/int/CreateProc: Trying makeproc evalf/int/ccquad: n = 2 integral estimate = 1.317562566127062387 n = 6 integral estimate = 1.338600467181040950 evalf/int/ccquad: n = 18 integral estimate = 1.338611019271054250 error = .1562023575177172825e-10 evalf/int/ccquad: n = 54 integral estimate = 1.338611019271379933 error = .1338611019271379933e-16 From ccquad, result = 1.338611019271379933 integrand evals = 55 error = .1338611019271379933e-16 tolerance = .6693055096356899665e-15

JCIxIVFyIz41aFE4ISM6

This time it used a non-NAG method called ccquad (Clenshaw-Curtis quadrature). You can get some information on these methods using Maple's help: ?evalf,int For the NAG methods, there's lots of information at the NAG web site . Maple uses some quite sophisticated methods for numerical approximation of integrals. We'll look at some rather less sophisticated ones, to get an idea of what is behind some of these. We'll start with some that you may remember from Math 101: left and right Riemann sums, the Midpoint Rule, Trapezoid Rule and Simpson's Rule.

restart; Suppose we want to approximate 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 numerically. The simplest way is called a Riemann sum: you divide the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= equal subintervals and approximate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== on each subinterval by its value at one point of the subinterval. 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 where 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. We'll usually have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= fixed, and look at different LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic='s, so I'll write everything as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The step size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the length of each subinterval. h := n -> (b-a)/n;

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsJkkiYkdGJSIiIkkiYUdGJSEiIkYsOSRGLkYlRiVGJQ==

The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic='th interval goes from 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 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImtGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==, where 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. X := (k,n) -> a + k*h(n); Calculus text authors seem to like "left sums", where you take the value at the left end of the subinterval, and "right sums", where you take the value at the right end. LeftSum:= n -> add(f(X(k-1,n))*h(n), k=1..n); eval(LeftSum(4), {a=0,b=1}); RightSum:= n -> add(f(X(k,n))*h(n), k=1..n); eval(RightSum(4), {a=0,b=1}); A better idea would be to go halfway along the subinterval. The Midpoint Rule approximation to the integral is 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. M := n -> add(f((X(k-1,n)+X(k,n))/2)*h(n), k=1..n); eval(M(4), {a=0,b=1}); M(n); The add function needs the range to be specified explicitly, not using a symbolic variable n. Here's a "formal" version of M. Mformal := n -> Sum(f((X(k-1,n)+X(k,n))/2)*h(n), k=1..n); eval(Mformal(n),{a=0,b=1}); For the Trapezoid Rule, we approximate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== on the subinterval by the average of its values at the two endpoints 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImtGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==. T := n -> add((f(X(k-1,n)) + f(X(k,n)))/2 * h(n), k=1..n); Tformal := n -> Sum((f(X(k-1,n)) + f(X(k,n)))/2 * h(n), k=1..n); Tformal(n); eval(T(4), {a=0, b=1}); Let's try a typical function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for which we know the true value of the integral, and look at the error (the difference between the true value and the approximation) for Midpoint and Trapezoid Rules with different values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. a := 0: b := 1: f := x -> 1/(x + 1): J := int(f(x),x=a..b); LeftSumErrors := [seq([n, evalf(J - LeftSum(n))], n = 1.. 20)]; RightSumErrors := [seq([n, evalf(J - RightSum(n))], n = 1.. 20)]; MidpointErrors := [seq([n, evalf(J - M(n))], n = 1 .. 20)]; TrapezoidErrors := [seq([n, evalf(J - T(n))], n = 1 .. 20)];

numer denom evalc gcdex JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=