Integrate this function. Break out the big guns.
Got two different constants when I asked for 1s!
You're the weirdest f I've ever met,
but you can bet before we're through,
mister, I'll make arctan out of u.
f(x) is 1 divided by
4+9x^2. Makes me want to cry!
If I can cancel out the 4 and 9,
Then I'm fine, here's what I'll do:
Somehow I'll make arctan out of u.
First cancel out the 4:
Divide it from the numerator.
Do the same on the bottom, so it's gone.
Pull the constant out the door,
Sayonara, see you later!
Now the 4's a 1, and so we can go on!
Chorus:
(Be arctan)
Don't hold your breath, you don't have to be scared.
(Be arctan)
One substitution and you will see:
(Be arctan)
We'll have du/(1+u^2),
from which we get arctan(u) + C.
Now this final process isn't so complex.
Just define your u as 1.5x.
So pull out the term 2/3,
Since du's 3dx/2.
Somehow I made arctan out of u!
Chorus:
(Be arctan)
Don't hold your breath, you don't have to be scared.
(Be arctan)
One substitution and you will see:
(Be arctan)
We'll have du/(1+u^2),
from which we get arctan(u) + C.
(Be arctan)
And then to finish steady your hand, sir.
(Be arctan)
Collect the terms and then we're free.
(Be arctan)
Resubstitute u to get the answer:
1/6 arctan(3/2 x) + C.