u-substitution problem?

evaluate the following integral by first making the substitution u^2=x+4:

integral: (dx)/(x*sqrt(x+4))

Stuck, need help! Thanks!

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  • cidyah
    Lv 7
    1 decade ago
    Favorite Answer

    ∫ dx / x sqrt(x+4)

    u^2 = x+4

    2udu = dx

    ∫ dx / x sqrt(x+4) = ∫ 2u du / (u^2-4)u

    = ∫ 2 du / (u^2-4) = 2 ∫ du/(u+2)(u-2)

    1/(u+2)(u-2) = A/(u+2) + B/(u-2)

    1 = A(u-2) + B(u+2)

    equate the coefficient of u

    0=A+B

    A=-B

    equate the constants

    1=-2A+2B

    1 = 2B+2B

    B=1/4

    A=-1/4

    2 ∫ du/(u+2)(u-2) =∫ [ -2/4(u+2)+2/4(u-2)] du

    = (-1/2) ln(u+2) + 1/2 ln(u-2)

    = (-1/2) ln (sqrt(x+4) + 2) + (1/2) ln(sqrt(x+4) -2) + C

  • 1 decade ago

    ∫dx/(x)√(x + 4)

    u = √(x + 4)

    u² - 4 = x

    2u du = dx

    ∫2u du/(u² - 4)(u)

    ∫2 du/(u² - 4)

    ∫2 du/(u + 2)(u - 2)

    1/2*∫(u + 2) - (u - 2)/(u + 2)(u - 2) du

    1/2*∫1/(u - 2) - 1/(u + 2) du

    1/2[ln|u - 2| - ln|u + 2|] + C

    1/2*ln|(u - 2)/(u + 2)| + C

    1/2*ln|(√(x + 4) - 2)/(√(x + 4) + 2)| + C

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