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In practice, it is useful to have a compact notation for the antiderivative What we do, instead of saying that the antiderivative of f (x) is F (x) + C, is to write f (x) dx = F (x) + C So, for example, cos x dx = sin x + C and x 3 + x dx = and e 2x dx = e 2x + C 2 x2 x4 + +C 4 2

CAMCO B-12345 (SAMPLE) FOLLOWER RADIUS FOLLOWER VELOC FOLLOWER ACCEL (INCHES) (INCHES/SEC) (INCH/SEC/SEC) 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750

2400m

RADIUS OF CURV (INCHES)

The symbol is called an integral sign (the symbol is in fact an elongated S ) and the symbol dx plays a traditional role to remind us what the variable is We call an expression like f (x) dx

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an inde nite integral The name comes from the fact that later on we will have a notion of definite integral that specifies what value C will take---so it is more definite in the answer that it gives

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CAMSHAFT ANGLE (DEGREES) PRESSURE ANGLE (DEGREES) 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 01562 06215 13864 24343 37414 52765 70014 88718 108388 128512 148579 168131 186955 204972 222122 238360 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13752 13756 13764 13777 13796 13821 13855 13897 13948 14008 14078 14157 14245 14343 14449 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00235 00937 02092 03676 05660 08005 10665 13588 16718 19993 23350 26728 30100 33459 36799 40111 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 169255 335217 494653 644462 781727 903776 1008235 1093069 1156628 1197674 1215410 1215410 1212119 1206203 1197674 1186552

We know that we must guess a trigonometric function Running through the choices, cosine seems like the best candidate The derivative of cos x is sin x So we immediately see that cos x is a better guess---its derivative is sin x But then we adjust our guess to F ( x) = cos( 3x + 1) to take into account the form of the argument This almost works: we may calculate that F ( x) = 3 sin( 3x + 1) We determine that we must adjust by a factor of 1/3 Now we can record our final answer as 1 sin( 3x + 1) dx = cos( 3x + 1) + C 3 We invite the reader to verify that the derivative of the answer on the righthand side gives sin( 3x + 1)

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FOLLOWER ANGLE (DEGREES)

We notice that the numerator of the fraction is nearly the derivative of the denominator Put in other words, if we were asked to integrate 2x +3

CAMCO B-12345 (SAMPLE) FOLLOWER RADIUS FOLLOWER VELOC FOLLOWER ACCEL (INCHES) (INCHES/SEC) (INCH/SEC/SEC)

RADIUS OF CURV (INCHES) 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 13750 19980 35933 152457 -75652 -32338 -21751 -17288 -15126 -14174 -14054 -14676 -15981 -17724 -20030 -23123 -27375

then we would see that we are integrating an expression of the form ( x) ( x) (which we in fact encountered among our differentiation rules in Section 25) As we know, expressions like this arise from differentiating log ( x) Returning to the original problem, we pose our initial guess as log[x 2 + 3] Differentiation of this expression gives the answer 2x/[x 2 + 3] This is close to what we want, but we must adjust by a factor of 1/2 We write our final answer as 1 x dx = log[x 2 + 3] + C 2 +3

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2800m

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