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Confused on what they substitued! answer given.

  1. Mar 21, 2007 #1
    Hello everyone i'm studying for my calc 3 exam and I'm lost how they went from this substitution, my professor did the work.

    Here is the problem and the answer under it:

    Well he showed the following:

    1/2* integral 0 to PI/2; integral 0 to 1 u*cos(uv) dv du;

    let t = uv;

    then he got:

    1/2 integral 0 to PI/2 u * 1/u sin(uv); now plug in 0 and 1 and he got:

    1/2 * integral 0 to PI/2 sin(u) du = 1/2 cos(u) and plug in PI/2 and 0 and u get 1/2.

    But what i'm lost on is, if you let t = uv;

    then if you took the derivative of t, with respect to v you would get 0 wouldn't you? or do i take the partial deriviatve? if i took the partial derivative with respect to v, then i would get t = u; which would make sense i think...maybe not
  2. jcsd
  3. Mar 21, 2007 #2


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    You have x-y=u and x+y=v.

    Ok, I'm so lost at how you wrote this stuff.

    Try writing integrals as bIa for integrating from b to a, for future reference. And show you du's and dv's because I'll get lost without those going around and other people would get lost too probably.
    Last edited: Mar 21, 2007
  4. Mar 21, 2007 #3
    right but then when you sub u in for x-y and v in for x + y your left with:
    u*cos(uv) dv du;

    and you have to make another substiution to do that integral, t = uv; which is where I get lost:
    t = (x-y)(x+y) ?
    t = x^2-y^2;
  5. Mar 21, 2007 #4
    You don't absolutely need a substitution to do that integral but if you can see it better by making one then if you have t=uv, then dt=udv. And you have the integral of cos(t)dt
  6. Mar 21, 2007 #5


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    No, it's not zero.

    dt/dv = u
  7. Mar 21, 2007 #6
    Thanks guys,

    now i have

    t = uv;
    dt/dv = u dv;

    cos(t) integral of that is sin(t), sub back in t = uv; and u get:
    sin(uv) now plug in 1 and 0 for v and your left with:
    integral 0 to PI/2 sin(u) du; but isn't the integral of sin(u) = -cos(u)?

    so wouldn't it be -1/2 as the naswer not 1/2?

    I understand u*1/u = 1; But when I did it, I didn't use that step at all to get sin(uv) after the subbing

    He had a step which I didn't quite understand..he said:

    1/2 integral 0 to PI/2 u*1/u sin(uv) du
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