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Math Help - Integral Questions

  1. #1
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    Integral Questions

    Can you help me.
    integral from 4 to 1
    x=( x^2-x+1)/(sqrt(x))

    integral from pie to pie/4
    x=secxtanx

    integral from 4 to x
    u=(2+(sqrt(u))^8


    integral
    x=sqrt(1+x^2x^5)


    integral
    x=tan^6xsec^4x
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    Quote Originally Posted by colaname View Post
    \int_1^4\frac{x^2-x+1}{\sqrt x}\,dx ...I'm assuming that when you say "from 4 to 1" you really mean from "1 to 4," but it doesn't make much difference.


    This one is easy. Just split up the fraction and use the properties of exponents to rewrite each term as a power of x.

    \int_{\pi/4}^\pi\sec x\tan x\,dx


    Well, what is the derivative of the secant? That should make the answer to this one obvious.

    \int_4^x(2+\sqrt u)^8\,du


    Substitute v=2+\sqrt u\Rightarrow dv=\frac{du}{2\sqrt u}, and then note that \sqrt u=v-2.

    \int\sqrt{1+x^2x^5}\,dx


    This has no elementary antiderivative. Did you write it correctly?

    \int\tan^6x\sec^4x\,dx
    Rewrite, using the identity \tan^2x+1=\sec^2x\colon

    \int\tan^6x\sec^4x\,dx

    =\int\tan^6x\sec^2x\sec^2x\,dx

    =\int\tan^6x(\tan^2x+1)\sec^2x\,dx

    =\int\tan^8x\sec^2x\,dx+\int\tan^6x\sec^2x\,dx.

    You should be able to take it from here.
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