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anonymous

  • one year ago

Find F ′(x) for F(x) = int from x cubed to 1 of (cos(t))^4

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  1. anonymous
    • one year ago
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  2. anonymous
    • one year ago
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    right so i know i have to use the fundamental theory of calculus but usually the end interval is an x so what should i do in this case ?

  3. campbell_st
    • one year ago
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    well isn't \[f(x) = \int\limits f'(x)~ dx\]

  4. ganeshie8
    • one year ago
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    use FTC + Chain rule

  5. anonymous
    • one year ago
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    + chain rule ?

  6. anonymous
    • one year ago
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    im sorry i dont follow , what do i apply the chain rule to, the cos(t^4) or the integral of cos(t^4)?

  7. ganeshie8
    • one year ago
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    Let \(G(t) = \int \cos(t^4)\,dt \) then do we have \(G'(t) = \cos(t^4)\) ?

  8. ganeshie8
    • one year ago
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    \[F(x) = \int\limits_{x^3}^1 \cos(t^4)\, dt = G(t) \Bigg|_{x^3}^1 = G(1) - G(x^3)\] now differentiate

  9. anonymous
    • one year ago
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    wait wait g(1) would equal \[\int\limits \cos(1)\] right?

  10. ganeshie8
    • one year ago
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    It doesn't matter, it is just a constat it vanishes when u differentiate

  11. ganeshie8
    • one year ago
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    \[\begin{align}F'(x) &= \dfrac{d}{dx} \left[G(1) - G(x^3)\right]\\~\\ &=0 - G'(x^3) * (x^3)'\\~\\ &=-\cos((x^3)^4)*3x^2 \end{align}\]

  12. anonymous
    • one year ago
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    okay i see

  13. ganeshie8
    • one year ago
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    do you see the chain rule part ?

  14. anonymous
    • one year ago
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    so \[-3x^2\cos (x^{12})\] , yes i do

  15. ganeshie8
    • one year ago
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    looks good! here is a general formula : \[\large \dfrac{d}{dx}\int\limits_{f(x)}^{g(x)}~h(t)\,dt ~~=~~ h(g(x))*g(x) - h(f(x))*f'(x)\]

  16. anonymous
    • one year ago
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    okay will take note of that thank you ! :D

  17. ganeshie8
    • one year ago
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    corrected a small mistake : \[\large \dfrac{d}{dx}\int\limits_{f(x)}^{g(x)}~h(t)\,dt ~~=~~ h(g(x))*g{\color{red}{'}}(x) - h(f(x))*f'(x)\]

  18. ganeshie8
    • one year ago
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    well, not small.. but you can see that the derivative kills the integral... you need to plugin the bounds and additionally you also need to multiply the derivative of the bounds

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