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Math Help - Integration question

  1. #1
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    Integration question

    Hey, i was wondering if anyone could help me with this question:

    integrate: 4x^3 exp(2x^4)

    i know what the answer is but i really dont understand how to do the question as apparently, it doesnt involve any integration at all!!!

    Thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by curlywurlysqurly View Post
    Hey, i was wondering if anyone could help me with this question:

    integrate: 4x^3 exp(2x^4)

    i know what the answer is but i really dont understand how to do the question as apparently, it doesnt involve any integration at all!!!

    Thanks
    a simple substitution of u = 2x^4 will do

    can you proceed?
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  3. #3
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    sorry, no! i still dont understand. I was told to differentiate the (2x^4) and then put the whole thing over the (8x^3), it then simplifies, and you dont do any integration.
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  4. #4
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    Just let:

    u=2x^{4}, \;\ du=8x^{3}, \;\ \frac{du}{8}=x^{3}

    Make the subs and it'll fall into place.
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  5. #5
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    i see what you mean kind of! but it isnt really working!!! please wouls someone help me go through all the steps.
    Thankyouuuu
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  6. #6
    TD!
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    <br />
\int {4x^3 e^{2x^4 } \,\mbox{d}x}  = \frac{1}{2}\int {\underbrace {e^{2x^4 } }_{e^u }\underbrace {8x^3 \,\mbox{d}x}_{\rm{d}u}}  = \frac{1}{2}\int {e^u \,\mbox{d}u} <br />
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by curlywurlysqurly View Post
    i see what you mean kind of! but it isnt really working!!! please wouls someone help me go through all the steps.
    Thankyouuuu
    fine, we will spoil you this time

    \int 4x^3 e^{2x^4}

    Let u = 2x^4

    \Rightarrow du = 8x^3 ~dx

    \Rightarrow \frac {du}2 = 4x^3 ~dx

    So our integral becomes:

    \frac 12 \int e^u~du = \frac 12 e^u + C

    = \frac 12 e^{2x^4} + C


    TD!'s underbraces make the substitution more explicit
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  8. #8
    Math Engineering Student
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    Quote Originally Posted by curlywurlysqurly View Post
    integrate: 4x^3 exp(2x^4)
    Note that (e^{2x^4})'=8x^3e^{2x^4}

    Integrate both sides e^{2x^4}+k_1=\int8x^3e^{2x^4}\,dx

    Divide by two and yields

    \frac12e^{2x^4}+k=\int4x^3e^{2x^4}\,dx
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  9. #9
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    Thumbs up

    Ooooohhhhh thankyou! very clever, now that you have done it i remember doin that at school! lol ahh well, needed that for my test tomorrow!!!
    Thanks!
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  10. #10
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    Sorry, that was actually to Jhevon but thanks to everyone!!
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