Reading the table of standard integrals off the table of derivatives

0:00Here is a list you are about to be handed. Thirteen rows. On the left of each one, a function. On the right, something whose rate of change is that function. It looks like thirteen new facts to learn. It is not. It is the derivative list you already own, with the arrow turned round. Read a row from right to left and it is a derivative you met long ago. Read the same row from left to right and it is an integral you have not.

0:33Same row. Same ink. The only thing that changed is which end you start from. So this is not a memorising problem. Turning the arrow round is free almost everywhere. Almost. There are three places on this list where it costs something, and those three places are the whole of the work. We are going to find all three by measuring, and not one of them by being told. First, the door everything here goes through.

1:05Nothing gets to be an answer because it looks like one. A function is an anti derivative of a rule when its measured rate of change is that rule. The difference quotient, taken from both sides of the input, narrowing until it settles, and then compared with the rule's own value at that input. Thirteen rows were put through that door. Each one written as two rules that know nothing about each other: one for the left column, one for the right.

1:38All thirteen passed, at five inputs each, inside the stretch each row lives on. The first row carries a second line, with no number on it, for the integrand that is just one. That passed too. Now the control that makes those thirteen passes worth having. Turn the sign of any answer over and put it through the same door. All thirteen fail. So the door can say no. Hold on to that. It is going to say no again.

2:12The first row is not one row. It is a family. Hand it any power of the input, and the answer is the next power up, divided by its own number. Forty one exponents were tried: every half step and every third of a step from minus five to five. Forty of them pass the door. One of them is refused. And here is the part worth watching: it is not refused by a condition somebody wrote in the margin.

2:44It is refused by the arithmetic. At an exponent of minus one, the next power up is the power nought, and its own number is nought. Nothing in this measurement divides by something that holds nought. So the answer does not come out wrong. It does not come out at all. The refusal belongs to the division, and the division is the same one every other row uses. So one over the input has no answer from the power rule.

3:17Which is exactly where most people stop, and conclude the case is missing. Let us find out instead. Five hundred and seventy four candidates were laid out: fourteen constants times forty one powers. Put that catalogue to a cube, and it comes back with a fourth power over four. One answer. Not two. Put it to one over the square, and it comes back with minus one over the input.

3:45Put it to a square root, and it comes back with two thirds of the three halves power. Now put it to one over the input. Nothing. Not one of the five hundred and seventy four. And yet the case is not missing. Ten rows further down the same list sits the answer, and it is not a power at all. It is a logarithm. That row passes the same door, at every input tried.

4:13The case was never lost. It was parked ten rows away from the row that excludes it, which is far enough that people stop looking. Six of the thirteen rows are trigonometric, and three of them carry a minus sign on the answer. Which three is not something to memorise either. It was measured. Each of the six named ratios was asked one question: is your own rate of change this row's integrand, or is it that integrand turned over?

4:47Three answered plus. The sine, the tangent, the secant. Three answered minus. The cosine, the cotangent, the cosecant. Three and three. And the three that answered minus are the three whose names begin with the same two letters. The co ones fall. That is the whole of the pattern, and it is one fact instead of six. Now the second place where turning the arrow round costs something. Two of the thirteen rows have the same function on the left.

5:22One over the square root of one less the square. And they have different functions on the right. One answers with the inverse sine. The other answers with minus the inverse cosine. Both of them passed the door. Before we look at what that means, one piece of housekeeping that decides whether any of this is a finding at all. Both of those inverse functions were found the same way, by halving.

5:51The inverse sine is the angle whose sine is the number you hand it, hunted down between minus and plus a quarter turn. The inverse cosine is the angle whose cosine is the number you hand it, hunted down between nought and a half turn. Two separate searches, on two separate stretches. Neither one is built out of the other. If I had defined the second as a quarter turn less the first, everything that follows would be a definition wearing the clothes of a discovery.

6:27So: one integrand, two answers, and the two answers are not equal. That reads as a misprint to anybody who has not yet believed that two anti derivatives can differ by a constant. Twenty eight inputs were handed to both of them, and their difference was read at each one. The difference takes exactly one value. At all twenty eight. And that one value is a quarter turn. So neither row is wrong. They differ by a constant, which is the only way two answers to the same integrand are allowed to differ.

7:06Here is the control, because one value is only impressive if some other pair gives more. Take a pair that is not two answers to one integrand: the inverse sine less the inverse tangent. At those same twenty eight inputs it takes twenty eight different values. One and twenty eight. That is the difference between a constant and a function, measured rather than asserted. And it happens a second time on this list, with the inverse tangent and minus the inverse cotangent: one value again, and a quarter turn again.

7:46The row that rescued the reciprocal is written with two upright bars around the input. It is tempting to read those bars as a convention. They are not. They are the difference between a row that is true and a row that asserts the logarithm of a negative number. Seventeen inputs a quarter apart, from minus two to two. With the bars in place, the row has a value at sixteen of them. Everywhere except nought.

8:17Rub the bars out, and it has a value at eight. Put both versions to four negative inputs. The row with the bars passes the rate door at all four. The row without them cannot even be asked the question. And the bars are earned rather than decreed: run the negative side on its own, as the logarithm of the input turned the right way up, and you get the same number the bars give, at all four inputs.

8:46Two cases, run separately, closing into one line. That missing point at nought now sends a bill, and this is the third place the reversal costs something. Take the row's own answer and add five to it, but only on the negative side of nought. Leave the positive side alone. That is a broken looking function. Put it through the door anyway, at seven inputs either side of the gap.

9:17It passes. Take three halves away on the negative side instead. That passes too. And those three are genuinely three different functions: at a negative input they take three different values, and at a positive input they take one value between them. So on a line with a point missing, the answer is not one constant's worth of freedom. It is two. Now look at one side alone. Eighty one candidate constants were laid out a quarter apart and asked which of them matches on the positive side.

9:55Exactly one does. The same one, whichever step was used on the far side. One unbroken stretch, one constant. A stretch with a hole in it, one constant per piece. That is why the stretch a formula lives on is not a formality, and it is the thing this list quietly leaves out. One row takes any base you like, raised to the input. Its answer is the same thing, divided by the logarithm of that base.

10:28Four bases were tried. All four pass. Try a base of one, and the row returns nothing at all, at every input. Again it is the division that refuses, because the logarithm of one is nought. Second exclusion, same cause as the first. Now the case people think needs its own rule. What if the base is the exponential's own? Then the divisor should become one, and the row should collapse into the exponential row above it.

11:00Rather than assert that, the base at which the divisor is exactly one was hunted down by halving. It came out at two point seven one eight two eight one, and onward, agreeing with the exponential's own value to twenty five places. So the general row does not need a separate rule for that base. At that base it already is the other row. Every row on this list is silent about the inputs it is allowed.

11:32That silence is normal, and it is recoverable. You just have to ask. Thirty three inputs an eighth apart, from minus two to two, handed to three of the integrands. The one with a square root of one less the square answers at fifteen of them. The fifteen strictly inside one either way. The one with one plus the square answers at all thirty three, because one plus a square never reaches nought.

12:02And the reciprocal answers at thirty two. Everywhere but nought. Then the part that makes this a measurement rather than a restatement: each row's answer goes silent at exactly the same inputs its integrand does. Fifteen and fifteen. Thirty three and thirty three. Thirty two and thirty two. The stretch is a property of the row, not of the column. And where a stretch ends at a point no fraction can land on, the silence shows up as growth instead.

12:36March in toward the quarter turn and the square of the secant passes a hundred, then ten thousand, then a million, and keeps going. Everything so far has been reading the list. Now the first thing that is not on it. Suppose you want an anti derivative for the cosine of twice the input. No row has that. But a row nearly has it. So guess the sine of twice the input, and then do the one step that turns a guess into a method: measure the guess.

13:11Its rate of change, over the integrand you wanted, at five inputs. Every one of the five comes back as two. One number, five times. So the guess is exactly twice too big, everywhere, and dividing it by two turns it into an answer that passes the door. Three integrands were done that way. The factors that turned up were two, one, and five. The one that came back as one needed no repair, which is the procedure telling you so rather than you noticing.

13:46Guess, measure, divide by whatever appeared, check. That is not a trick. It is four steps, and the third one is arithmetic. And the procedure has to be able to fail, or it is not telling you anything. Fourth integrand: the cosine of the square of the input. Guess the sine of the square. Same shape, same instinct. Measure it at five inputs, the same way. This time the five readings are five different numbers.

14:17Not one factor. Five. So no constant can repair this guess, and none of the fourteen constants in the catalogue does. That is worth more than it looks. The procedure did not quietly succeed on something it should not have. It told you the shape is wrong, and it told you by counting. Most integrands you meet are not on the list, and are not one guess away from it either.

14:47They are one rewrite away. Six of them were tested. A cube less one over a square. A two thirds power plus one. A three halves power with an exponential and a reciprocal. A sine plus a cosine. A cosecant times a cosecant and a cotangent together. And one less a sine, all over the square of a cosine. Each one was put to the whole list as it stands, written out to fifty five rows, against fourteen constants.

15:21Not one of the six is on it. Now split them. Term by term, or fraction by fraction. Thirteen pieces come out, and thirteen of the thirteen are on the list. The pieces add back up to the integrand they came from at every input tried, and all six of the answers pass the rate door. So rewriting is not preparation for the work. It is the work. The lookup afterwards takes no thought at all.

15:53The last of the six deserves its own look, because it is the one people reach for an identity on. One less the sine of the input, all over the square of the cosine. Nothing on the list matches that. Nothing is close. And the move is not an identity. It is a split. One over the square of the cosine, less the sine over the square of the cosine.

16:19The first piece is the square of the secant, which is a row. The second is the secant times the tangent, which is another row. Two rows, both already owned, and the answer is the tangent less the secant. Not every trigonometric integrand needs an identity. This one needed a division carried out. Here is one to do without being shown. A square root plus one over that same square root.

16:50Which is the half power plus the minus half power, and both of those are on the list. Four candidate answers were offered, and rather than work out which one is right, all four were put through the door at five positive inputs. Exactly one of the four passes. Two thirds of the three halves power, plus twice the square root. The other three fail. Which is the useful half of the result: a door that passes everything is not a door.

17:24One last thing, and it is the strangest item on the list. Thirteen rows come out of turning the derivative list round. Thirteen is what you have been shown. But a fourteenth row circulates alongside them, on revision sheets and in summaries. It takes one over one plus the square, and it answers with minus the inverse cotangent. Matching the two collections pair by pair, every one of the thirteen appears in the fourteen, and exactly one row of the fourteen appears nowhere in the thirteen. That one.

18:03So: is it wrong? No. It passes the same door every other row passes. And you already know why it is allowed, because you have seen this shape once before today. It is the second answer to a row that already has one, differing from it by a constant, exactly as the inverse sine and minus the inverse cosine differ. What is missing is not its truth. What is missing is its derivation. Nothing you have been shown ever produces it.

18:36So if you meet it cold on a revision sheet, it looks like a fourteenth fact. It is the tenth row, read a second way. Which leaves the last misconception, and it is the biggest one. That the list is closed. That an integrand not on it cannot be done. Six integrands were put to all fifty five rows and all fourteen constants. The tangent. The cotangent. The secant. The cosecant. One over four plus the square. And one over the root of four less the square.

19:13Not one of the six is on the list. Every one of the six has an answer, and all six of those answers pass the rate door. Minus the logarithm of the modulus of the cosine, for the tangent. The logarithm of the modulus of the sine, for the cotangent. And so on down. So what the list is short of is rows. It is not short of integrals. Which is the honest summary of the whole thing.

19:43Thirteen rows, none of them new, each one a derivative you already owned, read from the other end. One exponent the first row cannot take, and a row ten places away that takes it. One integrand with two answers, a quarter turn apart, which is the constant doing real work in front of you. And a silence about stretches that costs one extra constant every time a point goes missing.

20:12Learn the three costs, and the thirteen rows need no learning at all.