Factor out a 9.
Meaning,
1/(9+x^2) can be written as
1/9 * 1/(1+(x/3)^2)
Now let t=x/3
Hello, everyone!
A question:
Doesn't anyone teach the general formulas for inverse trig integrals?
. . . . . .du
. . ∫ ------------ . = . arcsin(u/a) + C
. . . .√aČ - uČ
. . . . . .du
. . ∫ ---------- . = . (1/a)·arctan(u/a) + C
. . . .uČ + aČ
. . . . . . .du
. . I ------------- . = . (1/a)·arcsec(u/a) + C
. . . .u√uČ - aČ
A half-century ago, my professor derived these formulas for us.
. . We have never had to factor-out-4 or factor-out-9.
[rant]
I've made this query at other math sites.
No one has ever thanked me for these streamlined formulas.
Instead, someone replies with something like:
"I'd rather have my students understand the concepts
. . rather than memorizing a list of formulas"
as if memorizing is a Bad Thing.
My first response has been:
If you've already memorized a formula with uČ + 1
. . why not expand it to include the "a"?
They usually reply with hints that I'm a formula-happy moron.
(Have they ever seen any of my posts?
Isn't Understanding the main feature of all my explanations?)
And then I must respond with something like:
If Understanding is the most important issue,
. . then we must not memorize the Quadratic Formula.
Instead, we make the student complete-the-square every time.
Then someone says, "What's wrong with completing-the-square?
I'd rather have them understand the procedure, than just memorize a formula."
Then I must explain the Philosophy of Learning.
When we learn a new fact, we in fact memorize it.
. . Yes, we "commit it to memory" for future use.
Then (and here's the important part) . . . we move on.
We add the formula/shortcut to our arsenal (in the war against our own ignorance)
. . and progress to bigger and better things.
And it is hoped that we retain the reasoning behind the formulas/shortcuts.
We do not draw a line in the sand and say, "This is far enough!
. . From now on, everything will be derived ... no more blind memorization!"
If Memorization is a misdemeanor, then 2 Ś 3 .= .6 is at least a felony.
. . After all, it can derived from: 3 + 3 .= .6
In fact, throw out the addition tables!
. . {◊ ◊ ◊} + {◊ ◊ ◊} .= .{◊ ◊ ◊ ◊ ◊ ◊}
. . . 1-2-3 . . . . 1-2-3 . - . . .1*2*3-4-5*6
We can always stop and "count apples", right?
[/rant]