Hello, Soroban,
Answer #1:
. Answer # 3
Answer #2:
. Answer # 3
Answer #4:
. Answer # 3
Greetings
EB
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 found four ways 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, radagast found a stunning fifth method.
And a day or so later, skeeter provided a surprising sixth method.
So I offer the challenge again (less arrogantly this time):
. . Is there a seventh method?
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Method #1: "Long Division"
We have: .
. .
Therefore: .
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Method #2:
Multiply top and bottom by
. .
Therefore: .
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Method #3: Substitution
Let
Substitute: .
Now there are two ways to proceed . . .
. . (A) Partial Fractions
. .
. . . .
. . (B) Complete the square
. .
. . . .
Note: the substitution leads to the same integral.
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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: .
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I have "boxed" the four different-looking solutions.
Your mission (should you decide to accept it) is to prove them equivalent.
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Method #5: radagast's method
We note that: .
Then: .
. . and we have: .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This is
We have the equation: .
Therefore: . . (Solution #1)
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Method #6: skeeter's method
Let
Substitute: .
. .
Therefore: . . (Solution #3)
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Now you are fully prepared to impress/amaze/terrify your teacher.
Well done, Earboth!
My first experience with these multiple-answer integrals was:
. . which my professor said could be integrated in two ways.
(1) Let
. . .Substitute: .
(2) Let
. . . Substitute: .
Then he proceeded to show us a third method:
(3)
And asked us to show that the answers are equivalent.
. . (I later found a fourth method.)
Later he gave us: .
And asked us to find at least three ways to integrate it . . .