8 Techniques of Integration
Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. For example, faced with x10 dx Z 11 10 11 10 we realize immediately that the derivative of x will supply an x :(x )′ = 11x . We don’t want the “11”, but constants are easy to alter, because differentiation “ignores” them in certain circumstances, so
d 1 1 x11 = 11x10 = x10. dx 11 11 From our knowledge of derivatives, we can immediately write down a number of an- tiderivatives. Here is a list of those most often used:
xn+1 xn dx = + C, if n = 1 n +1 6 − Z 1 x− dx = ln x + C | | Z ex dx = ex + C Z sin xdx = cos x + C − Z 163 164 Chapter 8 Techniques of Integration
cos xdx = sin x + C Z sec2 xdx = tan x + C Z sec x tan xdx = sec x + C Z 1 dx = arctan x + C 1 + x2 Z 1 dx = arcsin x + C √1 x2
Z −