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Thread: How to solve integral of high degree polynomial without expansion?

  1. #1
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    How to solve integral of high degree polynomial without expansion?

    How to solve integral of polynomial for example $$\int {(x^2 - 3x)}^5$$
    Can we solve this without expansion $$\int {(2x - 6)}^3 dx $$
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    Re: How to solve integral of high degree polynomial without expansion?

    the second one we can do using substitution

    $u = 2x-6$

    $du = 2 ~dx$

    $\displaystyle \int (2x-6)^3~dx \Rightarrow \int u^3~\dfrac{du}{2} = \dfrac{u^4}{8} = \dfrac{(2x-6)^4}{8} = 2(x-3)^4$

    One way or another you're going to have to expand the first one.
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    Re: How to solve integral of high degree polynomial without expansion?

    I see.. thanks. The answer is $$2x^4 - 24x^3 + 108x^2 - 216x + C$$
    And the expansion of $$(2x - 6)^4/8$$ is not like the answer.
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    Re: How to solve integral of high degree polynomial without expansion?

    For the second one, I would start by factoring out $2^3$: $8 \int (x- 3)^3 dx$. Let u= x- 3 so du= dx. The integral becomes $8 \int u^3 du= \frac{8}{4} u^4+ C= 2 (x- 3)^4+ C$.

    You say "the expansion of $(2x- 6)^4/8$ is not like the answer."

    The expansion is $2(x^4- 12x^3+ 54x^2- 108x+ 81)= 2x^4- 24x^3+ 108x^3- 216x+ 162$. What is "the answer"?
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    Re: How to solve integral of high degree polynomial without expansion?

    Ok thank you
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    Re: How to solve integral of high degree polynomial without expansion?

    Quote Originally Posted by Helly123 View Post
    How to solve integral of polynomial for example $$\int {(x^2 - 3x)}^5$$
    None answered the first one. If you know that: $(x^3-3x)^5=x^{15}-15x^{13}+90x^{11}-270x^9+405x^7-243x^5$
    wouldn't it be easy to find $\int {{{({x^3} - 3x)}^5}}~?$ See here.
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    Re: How to solve integral of high degree polynomial without expansion?

    Quote Originally Posted by Plato View Post
    None answered the first one. If you know that: $(x^3-3x)^5=x^{15}-15x^{13}+90x^{11}-270x^9+405x^7-243x^5$
    wouldn't it be easy to find $\int {{{({x^3} - 3x)}^5}}~?$ See here.
    the question did ask how to do it without expanding it
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    Re: How to solve integral of high degree polynomial without expansion?

    Quote Originally Posted by Plato View Post
    None answered the first one. If you know that: $(x^3-3x)^5=$
    The integrand is $(x^2 - 3x)^5$.
    Last edited by greg1313; Mar 2nd 2018 at 07:44 PM.
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    Re: How to solve integral of high degree polynomial without expansion?

    Quote Originally Posted by romsek View Post
    the question did ask how to do it without expanding it
    That is my whole point. Why not just expand it. Doing so really does simplify the task.
    With the tools available now a lot of topics that we thought were important are now obsolete. Partial fractions for example.
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