Results 1 to 2 of 2
Like Tree4Thanks
  • 4 Post By Prove It

Math Help - integration suitable substituition

  1. #1
    Member
    Joined
    Apr 2014
    From
    canada
    Posts
    124

    integration suitable substituition

    for this question, the question only stated SUITABLE substituition, what substituition should i use? this substituion does not involve trigo functions , am i right?integration suitable substituition-dsc_0143-2-1-.jpg
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,404
    Thanks
    1293

    Re: integration suitable substituition

    I'd write $\displaystyle \begin{align*} \int_0^4{\frac{1}{\left( 1 + \sqrt{x} \right) ^2}\,dx} &= \int_0^4{\frac{2\sqrt{x}}{2\sqrt{x}\left( 1 + \sqrt{x} \right) ^2}\,dx} \\ &= 2\int_0^4{ \frac{\sqrt{x}}{\left( 1 + \sqrt{x} \right) ^2} \left( \frac{1}{2\sqrt{x}} \right) \, dx } \end{align*}$

    So now make the substitution $\displaystyle \begin{align*} u = 1 + \sqrt{x} \implies du = \frac{1}{2\sqrt{x}}\,dx \end{align*}$ and note that $\displaystyle \begin{align*} u(0) = 1 \end{align*}$ and $\displaystyle \begin{align*} u(4) = 3 \end{align*}$, the integral becomes

    $\displaystyle \begin{align*} 2\int_0^4{\frac{\sqrt{x}}{\left( 1 + \sqrt{x} \right) ^2} \left( \frac{1}{2\sqrt{x}} \right) \, dx} &= 2\int_1^3{\frac{u - 1}{u^2}\,du} \\ &= 2\int_1^3{ u^{-1} - u^{-2}\,du } \\ &= 2 \left[ \ln{|u|} + u^{-1} \right] _1^3 \\ &= 2\left[ \left( \ln{|3|} + 3^{-1} \right) - \left( \ln{|1|} + 1^{-1} \right) \right] \\ &= 2 \left[ \ln{(3)} + \frac{1}{3} - 0 - 1 \right] \\ &= 2\left[ \ln{(3)} - \frac{2}{3} \right] \end{align*}$
    Thanks from Krahl, delso, topsquark and 1 others
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Suitable function H(H(x))=H(x)
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 10th 2011, 06:38 PM
  2. Suitable model?
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 25th 2008, 04:55 AM
  3. Help with substituition integration
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 6th 2008, 08:30 PM
  4. Replies: 6
    Last Post: January 17th 2008, 05:25 PM

Search Tags


/mathhelpforum @mathhelpforum