axis. We noticed that this didn’t give a very good approximation unlessthe number of rectangles was rather large. It should be clear from the rectangular representationof the area, that we could do better in estimating the area if we formed trapezoids using both sidesof the interval. This should give a better approximation.
5.7 - The natural logarithm - IntegrationObjectives1. Use the Log Rule for Integration to integrate a rational function2. Integrate Trigonometric FunctionsWe know how to differentiate a logarithm. We can also use the definition in a slightly modifiedform to integrate functions involving fractions:Log Rule:integraltext1xdx= ln|x|+CGeneral Log Rule:integraltext1udu= ln|u|+CSo we want to get used to sometimes lettingube the denominator of a fraction, and hope thatthe numerator gives us theduthat we need.Example:Findintegraldisplayxx2+ 1dxFindintegraldisplay3x+ 1dxFindintegraldisplay21-3xdx