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Math Help - [SOLVED] Evaluating Definite Integral (if exists)

  1. #1
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    [SOLVED] Evaluating Definite Integral (if exists)

    Hi,
    I'm I on the right track with this problem? Thanks in advance!

    1. The problem statement, all variables and given/known data
    Evaluate the definite Integral, if it exists


    2. Relevant equations
    integral (lower limit = 0, upper limit = square root of pi) of x cos (x^2) dx


    3. The attempt at a solution
    integral (lower limit = 0, upper limit = square root of pi) of x cos (x^2) dx

    Let U = x^2

    du/dx = 2x

    du = 2xdx

    dx = du / 2x

    integral (lower limit = 0, upper limit = square root of pi) x cos u du/2x

    (x/2 * - sin u 1/2x) + C

    -1/4 [ sin u ] + c

    -1/4 [ sin x^2 ]

    -1/4 [ sin pi ] - [ 1/4 sin 0 ]

    = 0
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by sparky View Post
    Hi,
    I'm I on the right track with this problem? Thanks in advance!

    1. The problem statement, all variables and given/known data
    Evaluate the definite Integral, if it exists


    2. Relevant equations
    integral (lower limit = 0, upper limit = square root of pi) of x cos (x^2) dx


    3. The attempt at a solution
    integral (lower limit = 0, upper limit = square root of pi) of x cos (x^2) dx

    Let U = x^2

    du/dx = 2x

    du = 2xdx

    dx = du / 2x

    integral (lower limit = 0, upper limit = square root of pi) x cos u du/2x = (1/2) cosu du.
    Integrating this gives (1/2) sin(u). Nou substitute u=x^2 and solve (1/2)sin(x^2) for the upper and lower limits.


    (x/2 * - sin u 1/2x) + C

    -1/4 [ sin u ] + c

    -1/4 [ sin x^2 ]

    -1/4 [ sin pi ] - [ 1/4 sin 0 ]

    = 0
    There is a small error. Look at the edited part and solve.
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  3. #3
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    \int_{0}^{\sqrt{\pi }} x*\cos (x^{2})(2xdx)=\frac{(\sin (x^2))}{2}=0-0=0

    You don't need a C with a definite integral and I am not sure where you obtained the 1/4
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  4. #4
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    Quote Originally Posted by dwsmith View Post
    \int_{0}^{\sqrt{\pi }} x*\cos (x^{2})(2xdx)=\frac{(\sin (x^2))}{2}=0-0=0

    You don't need a C with a definite integral and I am not sure where you obtained the 1/4
    Ok, thanks very much harish21 and dwsmith!
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