DLS
  • DLS
Find i1/i2
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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DLS
  • DLS
\[\LARGE I_1=29\int\limits_{0}^{1} (1-x^4)^7dx\] \[\LARGE I_2=4\int\limits_{0}^{1} (1-x^4)^6dx\]
DLS
  • DLS
@zepdrix @ganeshie8 @experimentX @amistre64
DLS
  • DLS
Attempt: Ill try to write i1 in terms of i2,anyone can post if they got better methods :) \[Applying~~integration~~by~~parts\] \[\large I_1=x(1-x^4)^7]_{0}^{1}+28\int\limits\limits\limits_{0}^{1}((1-x)^4.x^4)dx\]

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DLS
  • DLS
\[\large I_1=0+28\int\limits\limits\limits\limits_{0}^{1}((1-x)^4.x^4)dx\]
experimentX
  • experimentX
do you know a binomial expansion.
DLS
  • DLS
Correction \[\large I_1=0+28\int\limits\limits\limits\limits\limits_{0}^{1}((1-x^4)^6.x^4)dx\]
DLS
  • DLS
and yes
experimentX
  • experimentX
use binomial theorem to expand it and integrate term by term.
DLS
  • DLS
....complex
experimentX
  • experimentX
no complex ... this is simple. \[ 29\int\limits_{0}^{1} (1-x^4)^7dx = 29 \int_0^1 \sum_{k=0}^7 (-1)^k \binom{7}{k}x^{4k}dx = \sum_{k=0}^7 (-1)^k \binom{7}{k} \frac{1}{4k + 1}\]
experimentX
  • experimentX
add *29 to that last one ... and solve the other same way ... then just put up in your calculator and add up to find the sum.
experimentX
  • experimentX
internet is pretty slow here ... plug this thing into wolf and see if they are equal or not Integrate[(1 - x^4)^7, {x, 0, 1}] Sum[(-1)^k Binomial[7, k]/(4 k + 1), {k, 0, 7}]

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