Math question for those who want to help; post your answers in comments

I hope that regular readers will not feel shortchanged if I post math questions here. The few times I’ve brought up a math question, it stimulated many commenters with excellent explanations and good discussion among those who like this stuff.

That said, here’s my issue. I’m getting ready to take Calc 2 this fall, and I find that the algebra is still the hardest thing. I understand the calculus concepts just fine; it’s untangling the algebra that gets me, and I suspect gets most math students.

⌠ x/(x+1)^1/2 dx = ⌠ u-1/√u du Subsitute u = x+1, du = dx

= ⌠ (√u – 1/√u) du Rewrite integrand

– ⌠ (u^1/2 – u^-1/2) du Fractional powers

integrate each individually — got all this OK

⌠ 2/3u^3/2 -2u^1/2 +C Antiderivatives

2/3(x+1)^3/2 – 2(x+1)^1/2 + C Replace U by X=1 (so far so good, but it’s the next thing that I don’t understand)

2/3(x+1)^1/2(x-2)+C Factor out (x+1)^1/2 and simplify

For the life of me, I don’t see how factoring out x+1)^1/2 leaves you with 2/3(x+1)^1/2(x-2) Wouldn’t factoring out 1/2 power from 3/2 power leave you with a power of 3? Of course you have a power of 1 on (x-2). I see a negative sign and a 2 there, but how did the x get in front of it to make x-2? How did (x+1)^3/2 become (x+1)^1/2?

Thanks in advance to the commenters. The book I’m using is Briggs and Cochran, Calculus Early Transcendentals, page 261, and the Student Solutions manual. This is the best book and SS manual that I’ve found.


About mindweapon

A mind weapon riding along with Four Horsemen of the Apocalypse.
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11 Responses to Math question for those who want to help; post your answers in comments

  1. FD says:

    Let’s start at
    2/3(x+1)^3/2 – 2(x+1)^1/2 + C
    and rewrite it a little more clearly:
    2/3(x+1)(x+1)^1/2 – 2(x+1)^1/2 + C
    Then factor out the common term (this looks cleaner on paper, of course – extra parentheses for clarity):
    ((x+1)^1/2) * (2/3(x+1) – 2) + C
    Simplify second parentheses:
    ((x+1)^1/2) * (2/3*x + 2/3 – 2) + C
    Combine constants in second parentheses:
    ((x+1)^1/2) * (2/3*x – 4/3) + C
    Then factor out 2/3 from the second parentheses:
    (2/3)*((x+1)^1/2) *(x-2) + C

  2. Jasper says:

    The formatting on your equations make them somewhat difficult to follow. But in answer to your first question: exponents are added when terms are multiplied. For example, x^2 * x^3 = x^5 (not x^6). (If you’re skeptical, substitute “2” for “x” and solve). Similarly, when terms are divided (as by factoring), the exponents are subtracted. So, factoring x^3 out of x^5 would leave you with x^2. In your example, factoring a 1/2 power from a 3/2 power should leave you with a power of (3/2 – 1/2), or 1.

  3. jrackell says:

    2/3(x+1)^3/2 – 2(x+1)^1/2
    (x+1)^1/2 (2/3 (x+1) – 2 )
    (x+1)^1/2[ 2/3 x + 2/3 – 2 ] = (x+1)^1/2 (2/3x – 4/3)
    = (factor out 2/3)
    2/3(x+1)^1/2 (x-2)

  4. jrackell says:

    i think the above was the answer but maybe didn’t answer your question:

    if you have x^3/2 and you factor x^1/2 the exponents add, that is,
    for any numbers a and b, x^a * x^b = x^(a+b)

    [when factoring x^a from x^b the above ‘law’ is x^a x^(b-a) = x^b, which is really what you are doing:
    (x+1)^1/2 ((x+1)^ (3/2-1/2)) = (x+1)^1/2 (x+1)^1

    • mindweapon says:

      Thanks very much jrackell! Much appreciated!

      • jrackell says:

        You’re welcome. I remember myself taking calculus and wound up learning algebra.

        You know, I think an online WN** tutoring service would be a great idea. it could connect the generations and instill a certain loyalty from youngsters. Sort of defanging in their minds (kids, say) the idea of white advocacy as nothing to be ashamed of. Sort of Golden Dawnish/ Case Pound approach of emphasizing the social/community before the political.

  5. Ryu says:

    Maybe you’re being confused by all the junk going on. Just take it step by step.

  6. Skeeter says:

    Lately, I often solve a Rubik’s cube to keep my mind sharp instead. There’s a seven-step solution online in which you have to memorize long patterns for each step, the last one being 18-22 cube-turns longs. I credit my short-lived experience in fundamentalist Christian school as a child with my ability now to memorize long sequences. We had to memorize long Bible passages. True, it’s just rote memory and not understanding of ideas, but I think you have to first know about something before you can think something intelligent about it.

  7. Anon says:

    “Wouldn’t factoring out 1/2 power from 3/2 power leave you with a power of 3?”

    factoring is going to make the term smaller not bigger, for example x^2 + x factors to (x+1) * x. so you can see the inner x is smaller than x^2 because the extra x is outside the term.

  8. Mike says:

    I’m in Calc II right now and I never factored out a term for a solution of an integral before. Only time we did that was during Calc I with derivative (quotient rule), Yeah, when you have exponents with the same base and you multiple, you don’t actually multiple the exponents, you add the exponents. Furthermore, when you divide an exponent with the same base, you subtract.

    “Wouldn’t factoring out 1/2 power from 3/2 power leave you with a power of 3?”

    You’re multiplying when you should add. 1/2 * 3/2 = 6/2 or 3. That’s not right.

    When you multiple exponents you should add, x^1/2 * x^3/2 you get 1/2 – 3/2 = 4/2 or 2; the answer would be x^2
    And when you factor out you factor you subtract x^3/2 = x^1/2 * x^1; you get 1/2 + 2/2 = 3/2

    Calculus books I used, where Varberg Calculus for Calc 1 (hated that book) and Stewart Essential Calculus: Early Transcendental for Calc II.

  9. Sam Barber says:

    I believe that the ” Mass Man ” video is relevant here and I have posted it along with my thoughts on the calculus inquiry.

    We MWs seem like regular men who aren’t expected to do anything above our stations in life beyond paying bills, having meaningless hollow working class/middle class/upper class Western lives and meaningless hollow children with no soul, memory or identity.

    However, many of us in the MW crowd are studying the philosophy of dynamic numbers, shapes, and ratios in the form of physics, calculus, chemistry and other hard concrete disciplines. We are neither professional scholars nor amateur geeks. We are the men who have glimpsed the world beyond the television and have reached backwards into the past to find our identities as well as the path to our futures. Science resonates naturally within us and we find harmony in the world by understanding it as only WE can by observation, analysis and contemplation.

    The video attached is catching, profound and eye opening to we newcomers and old salts alike.

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