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Calculus

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Integrate (1/2)sin(x^(1/2))dx.

I've tried using u-substitution, with u=sin(x^(1/2)). du would then = ((1/2)x^(-1/2))(cos(x^(1/2))). As you can see, this only make the problem more complicated. I don't get what to do. Thank you in advance!

  • Calculus -

    when all else fails
    go to wolfram alpha (Just type that in your browser)
    in the box type:

    Integrate (1/2)sin(x^(1/2))

    after you see the answer, click on "show steps" in the box on the upper right.

  • Calculus -

    integral (sin(sqrt(x)))\/2 dx = sin(sqrt(x))-sqrt(x) cos(sqrt(x))+constant
    Possible intermediate steps:\n integral (sin(sqrt(x)))\/2 dx\nFactor out constants:\n = 1\/2 integral sin(sqrt(x)) dx\nFor the integrand sin(sqrt(x)), substitute u = sqrt(x) and du = 1\/(2 sqrt(x)) dx:\n = integral u sin(u) du\nFor the integrand u sin(u), integrate by parts, integral f dg = f g- integral g df, where \n f = u, dg = sin(u) du,\n df = du, g = -cos(u):\n = integral cos(u) du-u cos(u)\nThe integral of cos(u) is sin(u):\n = sin(u)-u cos(u)+constant\nSubstitute back for u = sqrt(x):\n = sin(sqrt(x))-sqrt(x) cos(sqrt(x))+constant

  • Calculus -

    Obviously it does not copy and paste very well so go to the original. Your substitution approach was in fact what was used.

  • Calculus -

    let u= sqrtx

    du= 1/2*1/sqrtx dx
    dx= 2u du
    INT 1/2 sin (sqrtx) dx
    INT 1/2 sin u 2u du
    int u sin u du=-u cos u + sin u

    then change it back to x
    -sqrtx cosSqrtx+sinsqrtx
    check: take the derivative...

    -1cossqrtx/2sqrx+ -sqrtx(-sinsqrtx)1/2sqrx +cossqurx/2sqrtx

    -1/2(-sinSqrtx)
    1/2 sinSqrtx so it checks.

  • Calculus -

    very cool. Thanks, both of you! Does wolfram alpha give steps for all integration problems?

  • Calculus -

    Pretty much so
    your other one would be
    integrate 1/(x ln x)

  • Calculus -

    awesome! Thanks Damon ;D

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