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16 January, 15:40

Integrate 1 / ((x^5) * sqrt (9*x^2-1)) ... ?

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  1. 16 January, 19:10
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    Let 9x^2-1 = y^2

    => 18xdx = 2ydy

    => ydy = 9xdx

    lower limit = sqrt (9*2/9 - 1) = sqrt (1) = 1

    upper limit = sqrt (9*4/9 - 1) = sqrt (3)

    Int. [sqrt (2) / 3,2/3] 1 / (x^5 (sqrt (9x^2-1)) dx

    = Int. [sqrt (2) / 3,2/3] xdx / (x^6 (sqrt (9x^2-1))

    = 81 * Int. [1, sqrt (3) ] ydy / ((y^2+1) ^3y)

    =81 * Int. [1, sqrt (3) ] dy / (y^2+1) ^3

    y=tanz

    dy = sec^2z dz

    =81*Int [pi/4, pi/3] cos^4 (z) dz

    =81/4*int [pi/4, pi/3] (1+cos (2z)) ^2 dz

    =81/4 * Int. [pi/4, pi/3] (1+2cos (2z) + cos^2 (2z)) dz

    =81/4 * (pi/3-pi/4) + 81/4 * (sin (2pi/3) - sin (pi/2)) + 81/8 * (pi/3-pi/4)

    + 81/32 * (sin (-pi/3) - sin (pi))

    =81 (pi/48+pi/96+1/4 * (sqrt (3) / 2 - 1) - 1/32 * sqrt (3) / 2)

    =81/32 * (pi+3sqrt (3) - 8)
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