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Math Help - Indefinite Integrals (Antiderivatives)

  1. #1
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    Indefinite Integrals (Antiderivatives)

    Hi, Kristenx2 again. Same take-home test.
    I missed the day we did antiderivatives, but is seems easy enough... I hope.
    My only issue is when it comes to the chain rule... Trig functions screw me up.
    I would just appreciate someone checking my work for the first, and possibly showing me how to do the second. I know I should use the quotient rule, but really.
    *Since I can't make that fancy antiderivative squiggle line, I will substitute with / *

    (a) /(3secxtanx+7sec^2x)dx
    =3/2sec^2x+7tanx+c

    (b) /(3x^3+5x^7)
    --------------------------- dx
    (square root of x)

    = ?
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  2. #2
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    Re: Indefinite Integrals (Antiderivatives)

    Quote Originally Posted by Kristenx2 View Post
    Hi, Kristenx2 again. Same take-home test.
    I missed the day we did antiderivatives, but is seems easy enough... I hope.
    My only issue is when it comes to the chain rule... Trig functions screw me up.
    I would just appreciate someone checking my work for the first, and possibly showing me how to do the second. I know I should use the quotient rule, but really.
    *Since I can't make that fancy antiderivative squiggle line, I will substitute with / *

    (a) /(3secxtanx+7sec^2x)dx
    =3/2sec^2x+7tanx+c

    (b) /(3x^3+5x^7)
    --------------------------- dx
    (square root of x)

    = ?
    Where does the chain rule come into this?

    For a) Differentiate sec(x)+tan(x). You should notice something which should make the integration easier.

    For b), we have:

    \int\frac{3x^3+5x^7}{\sqrt{x}}}dx

    =\int (\frac{3x^3}{\sqrt{x}}+\frac{5x^7}{\sqrt{x}}})dx

    =\int (3x^{\frac{5}{2}}+5x^{\frac{13}{2}})~dx
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  3. #3
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    Re: Indefinite Integrals (Antiderivatives)

    Oh! Thank you for your help! As you can see, I'm horribly confused, so I'm very unsure as to where anything comes in.

    I was looking on other sites as to how to do antiderivatives when there is a quotient, and they just said to flip it up.
    So I have
    /(3x^3+5x^7)(x^-1/2)dx.
    And ended up with
    6/7x^7/2+1/3x^15/2+c. I just used the power rule. I can use that even when there are constants, right?
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  4. #4
    Super Member Quacky's Avatar
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    Re: Indefinite Integrals (Antiderivatives)

    I think you really need a catch up session with your teacher. Also, for part b) I did most of the work for you. It should be \frac{6}{7}x^{\frac{7}{2}}+\frac{2}{3}x^\frac{15}{  2}+C Did you manage to get anywhere with part a)?
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    Re: Indefinite Integrals (Antiderivatives)

    Me, too. I noticed you got 2/3 where I got 1/3, that's probably just a mistake in my math.
    For the first one, would it just be
    3sec^2x+7tanx+c?
    Just leave the constants and figure out what makes the trig functions?
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  6. #6
    Super Member Quacky's Avatar
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    Re: Indefinite Integrals (Antiderivatives)

    No, did you try what I suggested in post #2?
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  7. #7
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    Re: Indefinite Integrals (Antiderivatives)

    Yes, I got sec(x)tan(x)+sec^2(x), which does look an awful lot like the original problem!
    However, how do I get the coefficients to come into play?
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  8. #8
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    Re: Indefinite Integrals (Antiderivatives)

    The coefficients don't change, as we have:

    \int(3sec(x)tan(x)+7sec^2(x))~dx

    \int{3}sec(x)tan(x)~dx+\int{7sec^2(x)}~dx

    =3\int{sec(x)tan(x)}~dx + 7\int{sec^2(x)}~dx
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  9. #9
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    Re: Indefinite Integrals (Antiderivatives)

    Quote Originally Posted by Kristenx2 View Post
    Hi, Kristenx2 again. Same take-home test.
    I missed the day we did antiderivatives, but is seems easy enough... I hope.
    My only issue is when it comes to the chain rule... Trig functions screw me up.
    I would just appreciate someone checking my work for the first, and possibly showing me how to do the second. I know I should use the quotient rule, but really.
    *Since I can't make that fancy antiderivative squiggle line, I will substitute with / *

    (a) /(3secxtanx+7sec^2x)dx
    =3/2sec^2x+7tanx+c

    (b) /(3x^3+5x^7)
    --------------------------- dx
    (square root of x)

    = ?
    This is just a matter of personal preference, but I don't like having to remember lots of integration formulas. For trigonometric functions I really only ever remember \displaystyle \begin{align*} \int{\cos{x}\,dx} = \sin{x} + C, \int{\sin{x}\,dx} = -\cos{x} + C \end{align*} and \displaystyle \begin{align*} \int{\sec^2{x}\,dx} = \tan{x} + C \end{align*} . Every other time I'll try to convert the integrand into some combination of these...

    \displaystyle \begin{align*} \int{3\sec{x}\tan{x} + 7\sec^2{x}\,dx} &= 3\int{\sec{x}\tan{x}\,dx} + 7\int{\sec^2{x}\,dx} \\ &= 3\int{\frac{1}{\cos{x}} \cdot \frac{\sin{x}}{\cos{x}}\,dx} + 7\tan{x} + C \\ &= 3\int{\cos^{-2}{x}\sin{x}\,dx} + 7\tan{x} + C \\ &= -3\int{\cos^{-2}{x}\left(-\sin{x}\right) dx} + 7\tan{x} + C \end{align*}

    Now make the substitution \displaystyle \begin{align*} u = \cos{x} \implies du = -\sin{x}\,dx \end{align*} and the integral becomes

    \displaystyle \begin{align*} -3\int{\cos^{-2}{x}\left(-\sin{x}\right) dx} + 7\tan{x} + C  &= -3\int{u^{-2}\,du} + 7\tan{x} + C \\ &= \frac{-3u^{-1}}{-1} + 7\tan{x} + C \\ &= \frac{3}{\cos{x}} + 7\tan{x} + C \\ &= 3\sec{x} + 7\tan{x} + C\end{align*}

    which is the same answer you would get using the other method as well, seeing as \displaystyle \begin{align*} \frac{d}{dx}\left(\sec{x}\right) = \sec{x}\tan{x} \end{align*} as you found before
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  10. #10
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    Re: Indefinite Integrals (Antiderivatives)

    Is a take-home test meant to be your own work?

    Thread closed. See rule #6: http://www.mathhelpforum.com/math-he...hp?do=vsarules.

    Thread closed.

    Quote Originally Posted by Kristenx2 View Post
    Hi, Kristenx2 again. Same take-home test.
    I missed the day we did antiderivatives, but is seems easy enough... I hope.
    My only issue is when it comes to the chain rule... Trig functions screw me up.
    I would just appreciate someone checking my work for the first, and possibly showing me how to do the second. I know I should use the quotient rule, but really.
    *Since I can't make that fancy antiderivative squiggle line, I will substitute with / *

    (a) /(3secxtanx+7sec^2x)dx
    =3/2sec^2x+7tanx+c

    (b) /(3x^3+5x^7)
    --------------------------- dx
    (square root of x)

    = ?
    Follow Math Help Forum on Facebook and Google+

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