By Stefan Hildebrandt
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Additional resources for Analysis 2 (Springer-Lehrbuch)
Notice that the x general solution of the differential equation has a constant of integration and represents a family of curves in the plane. The particular solution is one of these curves. Study Tips: fx" dx = -n+l- + C is not valid for n = -1. We do not yet know how to find the x"+! • The power rule anti derivative • f x dx . We will return to this question when we study logarithms. :.. Finding antiderivatives is more difficult than calculating derivatives, but remember that you can always check your answer to an integral question by differentiating the result.
Note that cos y is positive on the given interval and that we can replace sin y with x in the final step. )1- x 2 Up to this point in the course, y has been a function of the independent variable x. Now we will look at related rates problems in which x and yare functions of a third variable, t. As t varies, so do x and y. In the next example, involving a circle, the variables r (radius) and A (area) are changing with respect to t (time). Example 3: Related Rates A pebble is dropped into a calm pond, causing circular ripples.
I . Basic Integration Rules Based on Derivative Formulas: 1. fOdx=c. 2. fkdx=kx+C. 3. H/(x)±g(x)]dx = fl(x)dx± fg(x)dx. 4. f x"dx=--+C, n:;t:-l. n+l X"+l 6. fcos x dx sin x + C. fsinxdx = -cosx + C. 7. f sec 2 xdx = tan x + C. 5. = Summary: Antidifferentiation (or integration) is the inverse of differentiation. The anti derivative of I(x) = 3x 2 is G(x) = x 3 + C because the derivative of G is I. ©2010 The Teaching Company. 53 Example 1: Finding Antiderivatives fcos x dx = sin x + C f( X5 x6 + sinx +3)dx =6-cOSX +3x+ C You can check that these are correct by differentiating the answers.