DISCOVERING MATHEMATICS WITH MAPLEAn Interactive Exploration for Mathematicians, Engineers, and EconometriciansChapter 5. Derivative and IntegralWorksheet 5b. IntegrationThis worksheet is devoted to integration - Riemann integration, to be precise - of functions of a single variable. The variety of techniques offered by Maple for indefinite integration and numerical calculation of definite integrals is impressive; we shall pick a few lovely flowers in the luxuriant garden called Maple integration. We shall also consider examples of partial integration and of the substitution method. Finally we shall have a look at some definite integrals with limits of integration that are themselves functions of a single variable.Intuitive Perception of IntegrationRiemann integration of a bounded function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= : 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 on an interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ is about measuring the area bounded by the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis and two vertical lines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The area below the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis, is counted negative. Areas bordered by the graphs of functions can only be properly defined and computed if these functions can be approximated very accurately by step functions. Therefore, integration is often visualized as a process in which an ever increasing number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of rectangles is used to collectively cover the area in question more and more accurately as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= increases.The student package includes some procedures which demonstrate this intuitive perception of integration. Let us begin with defining a function and then load this package.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiYtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Ji1GLDYlUSJ4RidGNkY5LUY9Ni1RJyZyYXJyO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1GIzYqRistRiM2Iy1JJW1zdXBHRiQ2JUZYLUkjbW5HRiQ2JFEiNEYnRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY9Ni1RKCZtaW51cztGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGXXAtRiM2Ji1GY282JFEiMkYnRkAtRj02LVExJkludmlzaWJsZVRpbWVzO0YnRkBGQkZFRkdGSUZLRk1GT0ZobkZqbi1GIzYjLUZgbzYlRlhGYXBGZm9GKy1GPTYtUSIrRidGQEZCRkVGR0ZJRktGTUZPRlxwRl5wLUYjNiUtRmNvNiRRIjNGJ0ZARmRwRlhGW3FGYXBGK0YrRitGKw==QyQtSSV3aXRoRzYiNiNJKHN0dWRlbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=To get a first impression of this function, we quickly plot its graph over some suitable interval, say 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.LUklcGxvdEc2IjYkLUkiZkdGJDYjSSJ4R0YkL0YpOyEiIyIiIw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=A description of the process of integrating a function over an interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+, usually starts with the partitioning of this interval into a (finite) number of subintervals, followed by calculating the function's supremum and infimum over each subinterval, so that lower sums and upper sums can be built. If this partition is constantly refined by adding intermediate points in such a way that the maximum of the individual lengths of the subintervals tends to 0, then, provided the function is integrable over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+, the corresponding upper sum and lower sum converge to a common limit: the (Riemann) integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+.Instead of lower and upper sums, we could also take middle sums to approximate the integral. A middle sum - this is a particular choice of Riemann sum - determines the total area of the rectangles with height LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImlGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==), where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImlGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== is the midpoint of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-th interval. Clearly a middle sum will approximate the integral better as the number of subintervals grows. With ten subintervals of equal length on the interval 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, we get the following picture:LUkqbWlkZGxlYm94RzYiNiYtSSJmR0YkNiNJInhHRiQvRik7ISIjIiIjIiM1L0kmY29sb3JHRiRJJmJsYWNrR0YkLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The area covered by all ten rectangles isQyUtSSptaWRkbGVzdW1HNiI2JS1JImZHRiU2I0kieEdGJS9GKjshIiMiIiMiIzUiIiItSSZ2YWx1ZUdGJTYjSSIlR0YlLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Letting the number of subintervals (of equal length) increase from 10 to 200 with steps of 10, we see that with each step the value of the middle sum gets closer and closer to a limit value. We realize of course that this must be the value of the integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= over 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, or, in other words, the area of the region enclosed by the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis and the lines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiMkYnRjVGNUYrRjU= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 2.PkklTVN1bUc2IjcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkLUkmZXZhbGZHRig2Iy1JKm1pZGRsZXN1bUdGJDYlLUkiZkdGJDYjSSJ4R0YkL0YzOyEiIyIiIywkSSJuR0YkIiM1L0Y5OyIiIiIjPw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Also take a look at the following figure showing the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and the middle sum on 400 subintervals. The rectangles are no longer individually visible; the area below the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= appears to coincide with the combined area of all 400 rectangles.LUkqbWlkZGxlYm94RzYiNiYtSSJmR0YkNiNJInhHRiQvRik7ISIjIiIjIiQrJS9JJmNvbG9yR0YkSSZibGFja0dGJA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This rather inefficient way of calculating areas is only meant to illustrate the integration principle; fortunately there are much better ways available. The Fundamental Theorem of Calculus provides such a method.Ly1JJEludEdJKF9zeXNsaWJHNiI2JC0uSSJmR0YmNiNJInhHRiYvRiw7ISIjIiIjLCYtSSJGR0YmNiNGMCIiIi1GMzYjRi8hIiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a primitive of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= on the interval 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. For polynomials like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, primitives are quickly found.PkkiRkc2Ii1JKHVuYXBwbHlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JC1JJGludEdGJzYkLUkiZkdGJDYjSSJ4R0YkRjFGMQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Although there is hardly any need for in a trivial case like this, it is no trouble at all to quickly check the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and see that it coincides with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. This is of course the property that is most characteristic of a primitive function.LUkmZXZhbGJHJSpwcm90ZWN0ZWRHNiMvLUklZGlmZkdGJDYkLUkiRkc2IjYjSSJ4R0YsRi4tSSJmR0YsRi0=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Finally the Fundamental Theorem will give us the exact value of the integral.QyQvLUkkSW50R0koX3N5c2xpYkc2IjYkLS5JImZHRic2I0kieEdGJy9GLTshIiMiIiMsJi1JIkZHRic2I0YxIiIiLUY0NiNGMCEiIkY2L0kwbnVtZXJpY2FsfnZhbHVlRzYiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRyaHNHRic2I0kiJUdGJA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Assignment 1 of worksheet DerivA5b.mws takes another look at middle sums.Integration of Rational FunctionsWe have suggested before, and we repeat it here: Maple can integrate very complicated functions. It is well known that every rational function has a primitive. In the process of indefinite integration of a ratinal function, partial fraction decomposition plays an essential role. Let us consider how Maple accomplishes this.It might test your integration skills to their limits if you tried to integrate the following function by hand:QyQ+SSppbnRlZ3JhbmRHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUqJiwwKiQ5JCIiJyIiIiokRi8iIiYhIiMqJEYvIiIlIiIkKiRGL0Y3ISIlKiRGLyIiI0YzRi8hIiciIihGMUYxLCpGMkYxRjhGMUY6RjFGMUYxISIiRiVGJUYlRjE=Ly1JJEludEdJKF9zeXNsaWJHNiI2JC1JKmludGVncmFuZEdGJjYjSSJ4R0YmRistSSRpbnRHNiQlKnByb3RlY3RlZEdGJUYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=It seems almost equally difficult to verify the answer. Maple can help by differentiating the result.LUklZGlmZkclKnByb3RlY3RlZEc2JC1JJHJoc0dGJDYjSSIlRzYiSSJ4R0YqLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This still does not get us very far. It seems that Maple has applied partial fraction decomposition. Let us try to give the derivative a more or less normal appearance.LUknbm9ybWFsRyUqcHJvdGVjdGVkRzYkSSIlRzYiLkkpZXhwYW5kZWRHRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Here we see the original integrand again. We could force Maple to first apply partial fraction decomposition withLUkoY29udmVydEclKnByb3RlY3RlZEc2JS1JKmludGVncmFuZEc2IjYjSSJ4R0YoSShwYXJmcmFjR0YoRio=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Partial Integration and Integration by SubstitutionA commonly used integration technique is partial integration. The procedures intparts of the package student may help us to improve our working knowledge of this standard method.In the next lines both partial integration and substitution are used to integrate an elementary function; occasional explanations and comments are added.QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKHN0dWRlbnRHRiUhIiI=PkkpaW50ZWdyYWxHNiItSSRJbnRHSShfc3lzbGliR0YkNiQqJkkieEdGJCIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0YnNiNGKkYrRio=LUkpaW50cGFydHNHNiI2JEkpaW50ZWdyYWxHRiRJInhHRiQ=PkkrUHJpbWl0aXZlMUc2Ii1JJnZhbHVlR0koX3N5c2xpYkdGJDYjSSIlR0YkLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Verification shows that Maple's answer is correct, for instance byLUklZGlmZkclKnByb3RlY3RlZEc2JEkrUHJpbWl0aXZlMUc2IkkieEdGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Now try and complete assignment 2 of worksheet DerivA5b.mws, where the method of partial integration is applied to a similar indefinite integral.Another integration technique is the substitution method. For this purpose the student package has a procedure called changevar. We use it tl change the variable from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and then we backsubstitute to return to the variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5QyQtSSRJbnRHSShfc3lzbGliRzYiNiQqJkkieEdGJiIiIi1JJGxvZ0c2JCUqcHJvdGVjdGVkR0YlNiMsJiokRikiIiNGKkYqRipGKkYpRio=QyQtSSpjaGFuZ2V2YXJHNiI2JC8sJiokSSJ4R0YlIiIjIiIiRixGLEkidEdGJUkiJUdGJUYsQyQtSSlpbnRwYXJ0c0c2IjYkSSIlR0YlLUkkbG9nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJInRHRiUiIiI=QyQtSSZ2YWx1ZUdJKF9zeXNsaWJHNiI2I0kiJUdGJiIiIg==PkkrUHJpbWl0aXZlMkc2Ii1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJ0R0YkLCYqJEkieEdGJCIiIyIiIkYvRi9JIiVHRiQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Verify Maple's answer by clicking the right mouse button on the primitive in the output region and selecting the option Differentiate from the pop-up menu.Of course, Maple could have produced both primitives without any help from us.Assignment 3 of worksheet DerivA5b.mws is about symbolic integration and the necessity of keeping a watchful eye on the results Maple produces.Numerical IntegrationIn many situations we are only interested in the numerical value of a definite integral. It then makes sense to call numerical integration procedures directly, instead of in a roundabout way via symbolic integration. This is done by evalf(Int(...)) with capital I (or with 'evalf/int'). Now symbolic integration is avoided altogether, and as this is often in vain for 'difficult' functions anyway, it saves time as well. We give an example.Int(exp(-x^2)*cos(x),x=0..Pi)
= evalf(Int(exp(-x^2)*cos(x),x=0..Pi));JSFH[Maple Update: Here is the same line converted to 2d math input: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 Information on Maple's Integration MethodsTo get inside information on the integration methods Maple uses, the value of the Maple variable infolevel should be increased.QyQ+SStpbnRlZ3JhbmQyRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JJXNxcnRHRi82IzkkIiIiLCYqJEY2IiIjRjdGN0Y3ISIiRiVGJUYlRjs=PiZJKmluZm9sZXZlbEdJKF9zeXNsaWJHNiI2I0kkaW50RzYkJSpwcm90ZWN0ZWRHRiUiIiY=Ly1JJEludEdJKF9zeXNsaWJHNiI2JC1JK2ludGVncmFuZDJHRiY2I0kieEdGJkYrLUkkaW50RzYkJSpwcm90ZWN0ZWRHRiVGJw==JSFHMaple did find a primitive, but the expression returned is rather complicated.Let us try to unravel Maple's output a bit. The procedures used are stated with a brief explanation. We won't go into the precise meaning of the information provided here. The primitive produced by Maple is written as a finite sum of expressions in which the elliptic integral 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 can be distinguished, and with a summation variable running through the roots of the fourth degree polynomial 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.Let us reset the infolevel to 0. To make sure this rather involved expression is the correct primitive, we quickly check its derivative.QyQ+JkkqaW5mb2xldmVsR0koX3N5c2xpYkc2IjYjSSRpbnRHNiQlKnByb3RlY3RlZEdGJiIiISIiIg==LUknbm9ybWFsRyUqcHJvdGVjdGVkRzYkLUklZGlmZkdGJDYkLUkkcmhzR0YkNiNJIyUlRzYiSSJ4R0YtLkkpZXhwYW5kZWRHRi0=JSFHA Repeated IntegralTake a look at the following repeated integral (note the combinations of capital I and lower case i of Int and int ).Int(Int(x*y/(x^2 + y^2)^2,x=y..y^2),y=1..2) =Int(int(x*y/(x^2 + y^2)^2,x=y..y^2),y=1..2);Int(int(x*y/(x^2 + y^2)^2,x=y..y^2),y=1..2) = int(int(x*y/(x^2 + y^2)^2,x=y..y^2),y=1..2);[Maple Update: Here is the conversion to 2d math input:QyQvLUkkSW50R0koX3N5c2xpYkc2IjYkLUYlNiQqKEkieEdGJyIiIkkieUdGJ0YtLCYqJEYsIiIjRi0qJEYuRjFGLSEiIy9GLDtGLkYyL0YuO0YtRjEtRiU2JC1JJGludEc2JCUqcHJvdGVjdGVkR0YmRipGNkYtLy1JJEludEdJKF9zeXNsaWJHNiI2JC1JJGludEc2JCUqcHJvdGVjdGVkR0YlNiQqKEkieEdGJiIiIkkieUdGJkYvLCYqJEYuIiIjRi8qJEYwRjNGLyEiIy9GLjtGMEY0L0YwO0YvRjMtRilGJw==LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRyaHNHRiQ2I0kiJUc2Ig==JSFHThis integral is equal to the double integral over the (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=)-region bounded by the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the parabola LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE= and the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = 2. The next Maple code colours this region red. Incidently, for some reason which eludes us, we only get the correct picture if the plot order in the first argument of plots[display] is exactly as shown.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This concludes worksheet DerivW5b.mws.