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Math Help - Some integration problems

  1. #1
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    Some integration problems

    hi

    I am having a bit of trouble on the following integration problems and was wondering if anybody could help me out.

    1) Integrate: ( x^2 ) / sqrt(9- x^2 ) dx

    I think this one deals with trig substitution, but I am unsure of the exact method

    2) Integrate: dx/(x^2 * sqrt(1 + x^2 ))

    3) Integrate: x^2 / ( 25 + x ^2 ) dx

    with 2 & 3, I'm really not sure of the method to start them. any help is appreciated; thanks alot!
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ChaosBlue View Post
    1) Integrate: ( x^2 ) / sqrt(9- x^2 ) dx

    I think this one deals with trig substitution, but I am unsure of the exact method
    First let 3y = x. Then y = (1/3)x and dy = (1/3)dx so:
    Int[x^2/sqrt(9 - x^2)dx = Int[(3y)^2/sqrt(9 - (3y)^2)*3dy] = 9*Int[y^2/sqrt(1 - y^2)dy]

    Now let y = sin(t). Then dy = cos(t) dt. Thus:
    Int[x^2/sqrt(9 - x^2)dx = 9*Int[y^2/sqrt(1 - y^2)dy] = 9*Int[sin^2(t)/sqrt(1 - sin^2(t))*cos(t)dt]

    = 9*Int[sin^2(t)/cos(t)*cos(t)dt] = 9*Int[sin^2(t)dt]

    Now you take it from here.

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ChaosBlue View Post
    2) Integrate: dx/(x^2 * sqrt(1 + x^2 ))
    Same idea as 1) but with a different trig substitution:
    Let x = tan(t). Then dx = sec^2(t). Thus:
    Int[x^2/sqrt(1 + x^2)dx] = Int[tan^2(t)/sqrt(1 + tan^2(t))*sec^2(t)dt]

    Now, 1 + tan^2(t) = sec^2(t):

    Int[x^2/sqrt(1 + x^2)dx] = Int[tan^2(t)/sqrt(1 + tan^2(t))*sec^2(t)dt] = Int[tan^2(t)/sec(t)*sec^2(t)dt]

    = Int[sec(t)*tan^2(t)dt] = Int[sin^2(t)/cos^3(t) dt]

    Now you take it from here.

    -Dan
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ChaosBlue View Post
    3) Integrate: x^2 / ( 25 + x ^2 ) dx
    This is a mix between 1) and 2):
    First let 5y = x. Then x =(1/5)y and dx = (1/5)dy. Thus:
    Int[x^2/(25 + x^2) dx] = Int[(5y)^2/(25 + (5y)^2) * 5dy] = 5*Int[y^2/(1 + y^2) dy]

    Now let y = tan(t). Take it from here.

    -Dan
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  5. #5
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    Talking

    Quote Originally Posted by ChaosBlue View Post
    hi

    I am having a bit of trouble on the following integration problems and was wondering if anybody could help me out.

    1) Integrate: ( x^2 ) / sqrt(9- x^2 ) dx

    I think this one deals with trig substitution, but I am unsure of the exact method

    2) Integrate: dx/(x^2 * sqrt(1 + x^2 ))

    3) Integrate: x^2 / ( 25 + x ^2 ) dx

    with 2 & 3, I'm really not sure of the method to start them. any help is appreciated; thanks alot!

    Hello,

    Here are the answers. They are all in order just click on the links and follow the steps. I know that some of them seem rather long, but in most of the cases I am just arrowing over with my calculator!


    1. Integrate: ( x^2 ) / sqrt(9- x^2 ) dx

    a. http://item.slide.com/r/1/114/i/Jxlz...Ljm6JoXZpvx5_/
    b. http://item.slide.com/r/1/168/i/Uu3z...OiZevEggkYso1/
    c. http://item.slide.com/r/1/104/i/o92O...V8xbT3_HltsdQ/
    d. http://item.slide.com/r/1/27/i/yIm0M...UXOqjulE05xAI/
    e. http://item.slide.com/r/1/23/i/SoQPx...g8hvAKCWf9G9I/

    2. Integrate: dx/(x^2 * sqrt(1 + x^2 ))

    a. http://item.slide.com/r/1/73/i/5oJLP...KdNNK5OsRn_9i/
    b. http://item.slide.com/r/1/79/i/KfHZF...UzhT3uV-MbaS7/
    c. http://item.slide.com/r/1/93/i/V6afN...S8tEmt7eXPToE/
    d. http://item.slide.com/r/1/139/i/aVVZ...kby9vaZKgd2fW/
    f. http://item.slide.com/r/1/114/i/7byb...zlporHYxZ4dUq/
    g. http://item.slide.com/r/1/51/i/5TMiq...xjZX7002bZSDm/
    h. http://item.slide.com/r/1/125/i/TY_G...Ct4frI5KXyoa8/
    i. http://item.slide.com/r/1/11/i/-1jHG...LY7V7YCyGmkw1/
    j. http://item.slide.com/r/1/159/i/mlDT...ZpehScT-26Kmb/
    k. http://item.slide.com/r/1/49/i/pHBDW...hgFuu1qiODc1T/
    l. http://item.slide.com/r/1/32/i/V29Ji...G4P6KQBW2NQTa/
    m. http://item.slide.com/r/1/101/i/eQCg...F6dQJz4zg7Zlr/
    n. http://item.slide.com/r/1/68/i/MPTar...5oMpn_9YHMHcq/
    o. http://item.slide.com/r/1/14/i/Rkigt...Qo-vhs5hFxUer/
    p. http://item.slide.com/r/1/162/i/NE8a...h9Zj-XCdNhgyJ/
    q. http://item.slide.com/r/1/21/i/3Jfju...yj5yRRr3aljDI/
    r. http://item.slide.com/r/1/21/i/-nsRc...PKzZpy86gwC9P/

    2. Integrate: x^2 / ( 25 + x ^2 ) dx

    a. http://item.slide.com/r/1/151/i/A3f_...rdKR3g9yXColQ/
    b. http://item.slide.com/r/1/151/i/dlYk...1la8WyLu-DFLT/
    c. http://item.slide.com/r/1/168/i/ZAqK...m8WTt6KnrGD78/
    d. http://item.slide.com/r/1/152/i/4ZB9...wHDd3gN109GBc/

    There you go now were done!
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  6. #6
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    Hello, ChaosBlue!

    All of them deal with Trig Substitution . . .



    . . . . . x▓ dx
    1) . ∫ ---------
    . . . . √9 - x▓

    Let: x = 3Ěsinθ . . dx = 3ĚcosθĚdθ
    . . and the radical becomes: 3Ěcosθ

    . . . . . . . . . . .3Ěsin▓θ
    Substitute: . ∫ --------- (3ĚcosθĚdθ) . = . 3 ∫ sin▓θ dθ
    . . . . . . . . . . .3Ěcosθ

    Can you finish it now?




    . . . . . . . dx
    2) . ∫ --------------
    . . . . .x▓√1 + x▓

    Let x = tanθ . . dx = sec▓θĚdθ
    . . and the radical becomes: secθ

    . . . . . . . . . . . sec▓θĚdθ . . . . . . .cosθ
    Substitute: . ∫ ------------- . = . ∫ ------- . = . ∫ cscθĚcotθ dθ . . . etc.
    . . . . . . . . . . .tan▓θĚsecθ . . . . . .sin▓θ




    . . . . . .x▓ dx
    3) . ∫ ----------
    . . . . .25 + x▓

    Let: x = 5Ětanθ . . dx = 5Ěsec▓θĚdθ
    . . The denominator becomes: 25Ěsec▓θ

    . . . . . . . . . . . 25Ětan▓θ
    Substitute: . ∫ ------------ (5Ěsec▓θĚdθ) . = . 5 ∫ tan▓θ dθ . = . 5 ∫ (sec▓θ - 1) dθ
    . . . . . . . . . . . 25Ěsec▓θ

    Got it?

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