8.1: Prelude to Techniques of Integration

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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

\[\int x^{10}\,dx\]

we realize immediately that the derivative of $x^{11}$ will supply an $ x^{10}$: $(x^{11})'=11x^{10}$. We don't want the "11'', but constants are easy to alter, because differentiation "ignores'' them in certain circumstances, so

\[{d\over dx}{1\over 11}{x^{11}}={1\over 11}11{x^{10}}=x^{10}.\]

From our knowledge of derivatives, we can immediately write down a number of antiderivatives. Here is a list of those most often used:

\[\displaylines{ \int x^n\,dx={x^{n+1}\over n+1}+C, \quad\hbox{if $n\not=-1$}\cr \int x^{-1}\,dx = \ln |x|+C\cr \int e^x\,dx = e^x+C\cr \int \sin x\,dx = -\cos x+C\cr \int \cos x\,dx = \sin x+C\cr \int \sec^2 x\,dx = \tan x+C\cr \int \sec x\tan x\,dx = \sec x+C\cr \int {1\over1+x^2}\,dx = \arctan x+C\cr \int {1\over \sqrt{1-x^2}}\,dx = \arcsin x+C\cr }\]

Contributors

David Guichard (Whitman College)