Initiall value problems!

Question

please help me:

y'' = 2t, y'(0) = 0

and i need to use U subsitution for these:

antideriv. sin(11x)dx

antideriv(3x-12)^7 times 4dx

antideriv(4x 1)^.5 dx

antideriv x(x 3)^.5dx

Answer 1

For the first one:  you know y'' = 2t.  A little integration tells you, then, that y' = t^2 + C.  Since you know the value of y'(0), you can determine the value of the constant of integration, C.  Once you have that, you can integrate again and get y(t) to within a constant.

For your others:  think about what makes the integral harder to do.  For instance, I presume that you can integrate sin(x).  So what's hard about this one?  That "11" sitting there in front of the x.  So, get rid of it:

u = 11x

du = 11 dx

dx = du/11

So your integral becomes int(sin(u) du/11) = (1/11) int(sin(u) du)

and it's reduced to the thing you already know.

Go thou and do likewise on the others.