1. ## Partial Fraction

Q:

2. 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

3. Originally Posted by qbkr21
Q:
Here

4. ## Re:

Thanks Guys!

5. 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]

6. ## You don't need partial fraction

You don't need partial fraction. Just use the substitution rule.

Re:

8. ## The integral can be found as follows

The integral can be evaluated as follows:

9. Originally Posted by Soroban
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
That's exactly I was to write.

Thanks Soroban.

10. ## partial fraction

partial fraction