Irish girl

Equation:

http://i.imgur.com/WsByyjU.png

Im not sure how to isolate the constant K on the right side with respect to t. Any help is greatly appreciated!

Question:

http://i.imgur.com/bI497Fv.png

After separating the differential equation, would it be the integral of 2x on one side and the integral of 1 over everything else on the other side? Really stuck on this integral. what's the best approach? U-sub? integration by parts?

Thanks!

Question: A particle is moving along a straight line. For time t ≥ 0, the velocity of the particle is given by v(t) = 3t^2 −12.

Let b be an arbitrary number greater than 10. Find the total distance traveled by the particle from time

t = 0 to time t = b. Your answer should be an expression involving b.

To get distance you integrate from 0 to b, and the answer would be b^3 + 12b, but the answer is b^3 - 12b + 32. How you get the constant 32? all I know is that v(0) = -12.

Thanks!

I know what the answer is but I am not correct. I applied the formulas to integrate for x and for y, but my integration is wrong somewhere.

Exact question:

Find the center of mass of the region below the graph of f(x) = sin2

x, above the x-axis,

between x = 0 and x = π/2.

Original problem:

I solved part (a) and got pi/4. The answer to part (b) is ((pi^3)/8 + pi/2), but I am not getting this answer. Would anyone mind writing out how to solve part (b)?

The answer is (1/2ln(2) - pi/12). I've tried multiple times but my answer is not close to what it should be. Thank you so much!