Use the substitution
Hello, niyati!
This is definitely my favorite integral.
There are at least seven ways to integrate it.
Here are a few of them . . .
. .
Then: .
Multiply top and bottom by
. .
Therefore: .
Let
Substitute: .
Partial Fractions: .
. . .
. . . my favorite method
Let
. . and: .
Substitute: .
. . Hence: . .and .
Therefore: .
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Of course, these four answers are equivalent.
. . You should verify this for yourself.
But now you are prepared to surprise/impress/terrify your teacher.
Thank you! (Both red_dog and soroban)
I hated how I kind of had to use partial fraction decomposition on the substitution method (the only way we learned it...-_-) but I'm just wondering how I am suppose to intuitively know to do that. :P Perhaps practice, and hopefully a very similar problem on my exam.
Again, thank you both for your help.