Question

Which integrals / derivatives can we consider tabled?

For brevity's sake when you're doing more complex integrals, we just say "this integral is tabled" and put its actual value. Usually, it's something like an elementary function on its own.

But can we assume that when we're explaining this? I mean, for a person in the "Introduction to X" parts we can gather that they might appreciate the extra explanation but for someone dealing with more complicated topics stopping for explaining something like `inttan(x)dx` can be just a waste.

Thoughts?

Answer 1

Which integrals / derivatives can we consider tabled?

Explanation:

It tends to be the ones that are used very often that are stated as tabulated (like `int sin^2axdx = x/2 - (sin2ax)/(4a)` for Physical Chemistry), but there is actually an official list of tabulated integrals called The CRC Standard Mathematical Tables, CRC Press.

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The tabulated integrals are introduced in section 5.3.13 (pg. 420). I think you should search to see if the integral is there before assuming that it's there (though it's pretty comprehensive, so in all likelihood it's there).

For educational purposes, I think you should just perform the integral, or refer to somewhere that has manually done the integral before (maybe you've done it before already?).

However, if the integral itself is more like an aside (if it's not important to the end goal), then sure, I would state that it is tabulated and give the general result.

For trigonometric functions, for example, the integral is in section 5.4.16 (pg. 438).

`int tanaxdx = -1/alogcosax = 1/alogsecax`

... although, technically, it's supposed to be:

`int tanaxdx = -1/aln|cosax| = 1/aln|secax|`

The "log" is also a bit misleading (unless I'm not seeing somewhere that it defines "log" as the natural logarithm...), as you'd solve it like this:

`int tanaxdx`

`= int (secaxtanax)/(secax)dx`

Let:
`u = secax`
`du = asecaxtanaxdx`

`= 1/a int 1/udu`

`= 1/a ln|u|`

`= color(blue)(1/a ln|secax| + C) = color(blue)(-1/a ln|cosax| + C)`

So even if you know the integral is there, you might want to check it if you know how to do it.

Tabulated derivatives are introduced in section 5.1.2, and they start on section 5.1.3 (pg. 394).

But again, don't neglect or be afraid to check the derivative/integral yourself. For instance... The tables say:

`d/(dx)["arc"cscx] = -1/(xsqrt(x^2 - 1)`

but to be a bit pedantic, I'd say it should be:

`color(blue)(d/(dx)["arc"cscx] = -1/(|x|sqrt(x^2 - 1))`

which is what I was taught in college calculus.

Wolframalpha says it is `-1/(sqrt(1 - 1/(x^2))x^2)`, and if you modify that to match the blue answer, it is the same.

Since `sqrt(x^2) = pmx`, you would change the sign of the derivative when pulling an `x` into the square root as `x^2` if the remaining one is not `|x|`. Having the remaining one as `|x|` requires that the `x` you had pulled in was positive because evidently, `x^2 > 0`. Otherwise, the derivative is then `(pm)`, rather than specifically `(-)`.