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Math Help - Integration, which one is right? (I'm confused)

  1. #1
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    Integration, which one is right? (I'm confused)

    I'm now little bit confused. I have textbook that says:
    \int \! x \left( {x}^{2}+1 \right) ^{2}{dx}\
    =\frac{1}{6}\left(x ^{2}+1\right)^{3}\

    Ok, I'll do it my way and Integrate function by expand polynom first:
    \int \! x \left( {x}^{2}+1 \right) ^{2}{dx}\
    =\mathop{\rm  }\int \left(x ^{5}+2\mathop{\rm  }x ^{3}+x \right)\mathop{\rm  } dx
    =\mathop{\rm  }\frac{x ^{6}}{6}+\frac{x ^{4}}{2}+\frac{x ^{2}}{2}

    I get different integral function. let x=3 then first integral function gives result: 500/3=166,6666666... but seconds gives 333/2 = 166,5.

    So, which one is right?
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  2. #2
    Eater of Worlds
    galactus's Avatar
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    They differ by a constant C=1/6

    \frac{(x^{2}+1)^{3}}{6}=\frac{x^{6}}{6}+\frac{x^{4  }}{2}+\frac{x^{2}}{2}+\frac{1}{6}
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  3. #3
    Lord of certain Rings
    Isomorphism's Avatar
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    Quote Originally Posted by tabularasa View Post
    I'm now little bit confused. I have textbook that says:
    \int \! x \left( {x}^{2}+1 \right) ^{2}{dx}\
    =\frac{1}{6}\left(x ^{2}+1\right)^{3}\
    Never miss the constant of integration!
    If you want to know how the book did it, try the substitution u = x^2
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  4. #4
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    Oh, clever. Thanks guys!
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  5. #5
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    Quote Originally Posted by Isomorphism View Post
    Never miss the constant of integration!
    If you want to know how the book did it, try the substitution u = x^2
    Ok, I try to substitute u = x^2 Therefore I get:

    u = x^2
    x = u^(1/2)
    du = 2x
    dx = du/2

    ok, then function looks like:
    <br />
\int \! x \left( u+1 \right) ^{2}\frac{du}{2}\<br />

    or should it be:
    <br />
\int \! u^{\frac{1}{2}} \left( u+1 \right) ^{2}\frac{du}{2}\<br />

    Could you help me little bit on this..what next or is it already wrong?
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  6. #6
    Senior Member DivideBy0's Avatar
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    Quote Originally Posted by tabularasa View Post
    I'm now little bit confused. I have textbook that says:
    \int \! x \left( {x}^{2}+1 \right) ^{2}{dx}\
    Let u = x^2+1

    \,du = 2x\,dx\Rightarrow \,dx = \frac{\,du}{2x}

    So the integral becomes

    \int x\cdot u^2 \cdot \frac{\,du}{2x}

    =\int u^2 \cdot \frac{\,du}{2}

    =\frac{1}{2}\int u^2 \,du

    =\frac{1}{2} \left(\frac{u^3}{3}\right)+C

    =\frac{1}{2}\frac{(x^2+1)^3}{3}+C

    =\frac{1}{6}(x^2+1)^3+C
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  7. #7
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    Quote Originally Posted by DivideBy0 View Post

    \,du = 2x\,dx\Rightarrow \,dx = \frac{\,du}{2x}

    So the integral becomes

    \int x\cdot u^2 \cdot \frac{\,du}{2x}

    Thank you! This was the part i've totally forgot with substitution. X to the bottom Now I see the light too.
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  8. #8
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    I have one more question.
    I want to integrate function x*(x+1)^3 with substitution u = (x+1)^3

    How it continues?
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  9. #9
    GAMMA Mathematics
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    Quote Originally Posted by tabularasa View Post
    I have one more question.
    I want to integrate function x*(x+1)^3 with substitution u = (x+1)^3

    How it continues?
    Substitute: u = x+1. The integrand then becomes (u-1)u^3 which simplifies to u^4 - u^3. I think you can handle it from here.
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