PocketMatrix

Ok, my whole class and my teacher are all getting frustrated with this problem, and I just can't get an answer that works. Me and the teacher worked out a solution, but it didn't fit, then me and a classmate spent about an hour on it today and came up with an answer that almost works for x < 0, but not x > 0. We have to solve this beast, it was on the AP test. Here it is:

a(x) = 1 - arctan(e^x)
v(x) = ?

Here's some background information-
y = arctan(x) , x = tan(y)
The antiderivative of arctan(ax) = xtan(x) - (1/2a) ln[1+(a^2)(x^2)] + c
a(x) is acceleration and v(x) is velocity, so to solve for velocity, integrate the acceleration.
Good Luck!

The answer is C.

I know how to do it. You're going to have to convert arctan(e^x) into a MacLaurin series summation or a Binomial theorem power series summation and integrate that summation. To get the summation formula you're going to have to take the derivative of the function arctan(u), which doesn't effect the outcome of the summation, which will come out to be (1-u^2)^-1, and then form the summation based on that. It will be something along the lines of:

x - integral of Sigma n=1 to Infinity of -1 choose n times (u^2)^n.

That's using the binomial theorem. You'd have to calculate out the choose and come up with a tighter Sigma notation before you integrate.

It's a very difficult Calc2 level question. Search for Taylor Series, MacLaurin Series, and the Binomial Theorem.