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What next after this [sry another c4 question] v.revolution watch

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    limits are 3,0

    π ∫ √[x/[x+1]]² ∂x

    π ∫ [x/[x+1] ∂x

    im confused what to do now, cause the c4 markscheme [paper e, last question] comes out with some method i don't understand.

    they just take 1 on the top and add 1 :confused: what is the point in this as it just cancels out?
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    (Original post by nas7232)
    limits are 3,0
    π ∫ [x/[x+1] ∂x
    Firstly you probably shouldn't use the partial derivative symbol, stick to the more appropriate d
    Adding and subtracting one gives the same result as if you were to use algebraic long division - it enables you to make the fraction proper so it is much easier to integrate.
    \pi \int_0^3 \frac{x}{x+1} dx \\

= \pi \int_0^3 \frac{(x+1)-1}{x+1} dx \\

= \pi \int_0^3 1 - \frac{1}{x+1} dx \\

= \pi [x-ln(x+1)]_0^1 \\

= \pi [1-ln2]
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    (x+1-1)/(x+1) = (x+1)/(x+1) - 1/(x+1) = 1 - 1/(x+1)

    They did that to simplify the fraction into something you can integrate easily now. Another way to do this would be by using long division.

    If you're not comfortable with this, then use the substitution u=x+1 with du=dx, this transforms your integral into:
    ∫ (u-1)/u du = ∫ u/u - 1/u du = ∫ 1 - 1/u du

    And btw, you shouldn't use ∂x to represent the 'dx' in ∫ dx.
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    ah i get it, but how do we know when to use this method?

    issit when there is a/(1+a)?

    Also, which other unwritten rules do we need to know for volumes of revolution or is this the only one? [At A2 C4 edexcel level]
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    (Original post by nas7232)
    ah i get it, but how do we know when to use this method?
    issit when there is a/(1+a)?
    Basically it's often advantageous to use methods such as this or long division when the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. It can also be useful in situations such as
    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    \int \frac{x^{2}}{\sqrt{x^{2}+1}
    but that's beyond the C4 syllabus.

    Also, which other unwritten rules do we need to know for volumes of revolution or is this the only one? [At A2 C4 edexcel level]
    The only rules you need to really remember for volume is V=\pi \int_{x_{a}}^{x^{b}} y^{2} dx and V=\pi \int_{y_{a}}^{y_{b}} x^{2} dy
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    Basically, whenever you have:
    \int \frac{x}{x \pm k} dx

    Consider adding and substracting k -- your goal is to create a fraction in the numerator that can cancel out the denominator and leave you with a simpler fraction.

    e.g. if it was x/(x-2), then: (x-2+2)/(x-2) = 1 + 2/(x-2).

    Or just use long division.
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    ok nice1

    so basically when x is on the top and something like x+k something on the bottom or ≡ equation
 
 
 
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