The Natural Logarithmic Function and Integration 5.2 The Natural Logarithmic Function and Integration Log Rule for Integration

Ex. Let u = 2x - 1 Ex. du = 2 dx

Ex. Find the area of the region bounded by the graph of , the x-axis, and the line x = 3. .5 Let u = x2 + 1 du = 2x dx 3

Recognizing Always ask yourself, is the top the derivative of the bottom. What does the numerator need to become the derivative of the denominator? It needs to be multiplied by 2.

Using long division before integrating. If a rational function has a numerator of degree greater than or equal to that of the denominator, division may reveal a form to which we can apply the Log Rule. Long division gives us…

Change of Variables with the Log Rule Let u = x + 1 du = dx and x = u - 1

Evaluate Divide top and bottom by x and you’ll get a u’/u. Day 1 stop Integrals of 6 basic trigonometric functions.

Evaluate

The electromotive force E of a particular electrical circuit is given by E = 3 sin 2t where E is measured in volts and t is measured in seconds. Find the average value of E as t ranges from 0 to .5 seconds. Average Value is given by Let u = 2t du = 2 dt

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