Hello, everyone!

Try this integral: .

I posted this problem at SOSMath three years ago.

I said (quite smugly as you will see):

. . . . ."So far, I've foundfourways to integrate it.

. . . . . And the four answers all look different.

. . . . . Give it a try. .I dare you to come up with a fifth method."

Within three hours,radagastfound a stunning fifth method.

And a day or so later,skeeterprovided a surprising sixth method.

So I offer the challenge again (less arrogantly this time):

. . Is there a seventh method?

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Method #1: "Long Division"

We have: .

. .

Therefore: .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Method #2:

Multiply top and bottom by

. .

Therefore: .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Method #3: Substitution

Let

Substitute: .

Now there aretwoways to proceed . . .

. .(A) Partial Fractions

. .

. . . .

. .(B) Complete the square

. .

. . . .

Note: the substitution leads to the same integral.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Method #4: Trig Substitution . . . my personal favorite.

Let

The denominator is: .

Substitute: .

Since . then angle is in a right triangle with and .

. . Hence: . .and .

Therefore: .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I have "boxed" the four different-looking solutions.

Your mission (should you decide to accept it) is to prove them equivalent.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Method #5:radagast's method

We note that: .

Then: .

. . and we have: .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This is

We have the equation: .

Therefore: . . (Solution #1)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Method #6:skeeter's method

Let

Substitute: .

. .

Therefore: . . (Solution #3)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Now you are fully prepared to impress/amaze/terrify your teacher.