Question:

Find the integral? ∫ dx / ( 9 + 2x²)x3/2?

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∫ dx / ( 9 + 2x²)3/2

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  1. (2/3) ∫ 1/(9+2x²) dx = (1/3) ∫ 1/(9/2 + x²) dx

    Recall that d/dx atan(x) = 1/(1+x²)

    Let au = x ; a du = dx

    ∫ 1/(a² + x²) dx = ∫ adu/(a² + a²u²) = (1/a ) ∫ du / (1 + u²)  = 1/a atan(u) + c

    ∫ 1/(a² + x²) dx = 1/a atan(x/a) + c

    (1/3) ∫ 1/(9/2 + x²) dx = (1/3) √2 / 3 atan (x √2/3 ) + C = √2/9 atan(x √2/3 ) + C

    Answer: √2/9 atan(x √2/3 ) + C


  2. quite interesting....

    ∫ dx / (9 + 2x²)^(3/2) =

    ∫ dx / √(9 + 2x²)³ =

    ∫ dx / [(9 + 2x²)√(9 + 2x²)] =

    let x = (3/√2) tan u →

    dx =  (3/√2) sec²u du

    thus, substituting, you get:

    ∫ dx / [(9 + 2x²)√(9 + 2x²)] = ∫ (3/√2)sec²u du/ {9 + 2[(3/√2)tan u]²}√{9 + 2[(3/√2)tan u]²} =

    ∫ (3/√2) sec²u du / {9 + 2[(9/2) tan²u]}√{9 + 2[(9/2) tan²u]} =

    ∫ (3/√2) sec²u du / [(9 + 9 tan²u)√(9 + 9 tan²u)] =

    factoring out 9:

    ∫ (3/√2) sec²u du / {9(1 + tan²u)√[9(1 + tan²u)]} =

    ∫ (3/√2) sec²u du / [9(1 + tan²u) 3√(1 + tan²u)] =

    according to trig identity, replace 1 + tan²u with sec²u:

    ∫ (3/√2) sec²u du / [27sec²u √(sec²u)] =

    ∫ (3/√2) sec²u du / [27sec²u (sec u)] =

    3 sec²u canceling out,

    ∫ (1/√2) du / (9 sec u) =

    taking the constants out,

    [1/(9√2)] ∫  (1 / sec u) du =

    [1/(9√2)] ∫ cos u du =

    [1/(9√2)] sin u + C

    to sum up:

    x = (3/√2) tan u → tan u = (x√2)/3

    then, finding out sin u in terms of tan u, you get:

    1/sin²u = csc²u = 1 + cot²u = 1 + 1/tan²u = [(tan²u + 1)/tan²u]

    thus

    sin²u =  [tan²u/(tan²u + 1)] → |sin u| = |tan u| /√(tan²u + 1) →

    sin u = tan u /√(tan²u + 1) →

    therefore, being tan u = (x√2)/3,

    sin u = [(x√2)/3] /√ {[(x√2)/3]² + 1} = [(x√2)/3] /√ [(2x²/9) + 1] =

    [(x√2)/3] /√[(2x²+ 9)/ 9] = [(x√2)/3] [3/√(2x²+ 9)] = [(x√2)/√(2x²+ 9)]

    and finally substituting back, you get:

    ∫ dx / (9 + 2x²)^(3/2) = [1/(9√2)] [(x√2)/√(2x²+ 9)] + C =  (1/9)x (2x²+ 9)^(-1/2) + C

    I hope it helps..

    Bye!

  3. i take it you are intergrating with respect to x;

    =int.3(9+2x^2)/2  dx

    =int.27+6x^2  dx

    =27x+(6/3)x^3 + c

    =27x+2x^3+c

    hope this helps

  4. that's tough. math is one of my weakest subjects.

  5. do your own homework duh

    answer is = you might be a floogle stoofle  

  6. i have no clue  

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