Lecture 2 Antiderivatives and Integration - W15 MAT21B LECTURE 2 ANTIDERIVATIVES BEGINNING INTEGRATION GEORGE MOSSESSIAN By now you know how to take

Lecture 2 Antiderivatives and Integration - W15 MAT21B...

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W15 MAT21B LECTURE 2: ANTIDERIVATIVES, BEGINNING INTEGRATION GEORGE MOSSESSIAN By now, you know how to take derivatives of functions, and it stands to reason that if you can take a derivative, you should be able to undo the taking of a derivative as well. This is called taking an antideriva- tive . Notice the word an , as opposed to the , which suggests that there may be more than one. Let’s see why this is true. Consider y = 2 x . Is there a function F ( x ) such that F 0 ( x ) = 2 x ? Of course, you say, F ( x ) = x 2 . Except that x 2 + 1 also works, as does x 2 + C for any real number C . Thus, there are actually an infinite number of antiderivatives of 2 x , but they all 1 look like x 2 + C . Another example: y = cos x has antiderivative sin x + C , y = sin x has antiderivative - cos x + C , and y = 2 x + sin x has antiderivative x 2 - cos x + C . In general, the antiderivative of x n is 1 n + 1 x n +1 + C . You can check this by taking the derivative of that result: 1 n + 1 x n +1 + C 0 = 1 n + 1 · ( n + 1) x n + 0 = x n . When there are constants involved, you have to “un-chain” them, that is, make sure they get cancelled out in the chain rule that follows from taking the derivative, like so: (sin kx ) 0 = - 1 k cos( kx ) , so that when you take the derivative of - 1 k cos kx , the k which results from chain rule on kx cancels out with the 1 k . This leads to a sort of general philosophy of finding antiderivatives: first, make an educated guess which may or may not have the correct constant coefficients, and then adjust for those. For example: if I want to

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