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Integration of a function of x

  1. Nov 25, 2016 #1
    1. The problem statement, all variables and given/known data
    ##∫45.1/3x^2 (4-2x)^3dx##



    2. Relevant equations


    3. The attempt at a solution

    ##45/3∫x^2(4-2x)^3dx = let u = x^2 du= 2x, dv= (4-2x)^3 v=(2-x)/-4 ##

    using intergration by parts is this right
     
  2. jcsd
  3. Nov 25, 2016 #2

    hilbert2

    User Avatar
    Science Advisor
    Gold Member

    It's not obvious whether you mean ##\frac{45.1}{3x^2 (4 - 2x) ^3}## or ##\frac{45.1}{3}x^2 (4-2x)^3##.
     
  4. Nov 25, 2016 #3

    fresh_42

    Staff: Mentor

    If you want to integrate ##\int \frac{45}{3} x^2(4-2x)^3dx=15 \int x^2(4-2x)^3dx## by parts, you will have to apply it three times and in the end, the polynomial has to be of degree ##6##, for you started with a degree ##5## polynomial. I'm rather sure that performing the multiplication ##x^2(4-2x)^3## and integrating term by term is almost faster.

    The formula by parts goes ##\int u'v = uv - \int uv'##
     
  5. Nov 25, 2016 #4

    CAF123

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    Gold Member

    If you are solving ##\int x^2 (4-2x)^3 dx## then integration by parts is perhaps overkill. You could simply expand out all terms or first make the substitution ##u = 4-2x## which means you no longer have to deal with expanding out a cubic power. If you have to use integration by parts, your integration of v is incorrect.
     
  6. Nov 25, 2016 #5
    sorry i mean the latter...........
     
  7. Nov 25, 2016 #6
    I expanded and multiplied out the factors...and got it as follows,

    ##45/3∫x^2(16-16x+4x^2)(4-2x) = 45/3∫x^2(64-32x-64x+32x^2+16x^2-8x^3)dx = 45/3∫(-8x^5+48x^4-96x^3+64x^2)dx##

    this is an easy integral i was just tired, i will try use the substitution method and see what comes out.
    now on using substitution
    let ##u=4-2x, →dx=du/-2, x= (4-u)/2 ⇒x^2=(16-8u+u^2)/4,

    ⇒45/3∫((u^2-8u+16)/4)). (u^3) (du/-2) = 45/-24∫(u^5-8u^4+16u^3)du ##

    which will give correct solution as the other long method.
    Thank you guys, greetings from Africa.
     
    Last edited: Nov 26, 2016
  8. Nov 28, 2016 #7

    Mark44

    Staff: Mentor

    @chwala, please post questions on calculus in the Calculus & Beyond section, not the Precalculus section, where you originally posted this thread.
     
  9. Nov 29, 2016 #8
    OK sir
     
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