Results 1 to 4 of 4

Math Help - integration problem

  1. #1
    Junior Member
    Joined
    Feb 2008
    Posts
    63

    integration problem

    x^3*(x^2+1)^(1/2)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    \int{x^{3}\sqrt{x^{2}+1}}dx

    If you let u=x^{2}+1, \;\ \frac{du}{2}=xdx \;\ u-1=x^{2}

    it'll turn it to something much easier.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    13
    To get rid of the square root, one also can set u^2=x^2+1.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,735
    Thanks
    642
    Hello, Andreamet!

    \int x^3(x^2+1)^{\frac{1}{2}}\,dx
    We can do it By Parts . . .

    . . . \begin{array}{ccccccc}u & = & x^2 & & dv & = & x(x^2+1)^{\frac{1}{2}}\,dx \\ du &=& 2x\,dx & & v &=&\frac{1}{3}(x^2+1)^{\frac{3}{2}} \end{array}

    And we have: . \frac{1}{3}x^2(x^2+1)^{\frac{3}{2}} - \frac{2}{3}\int x(x^2+1)^{\frac{3}{2}}\,dx\quad \hdots\quad etc.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Another approach is Trig Substitution . . .

    Let x \:=\:\tan\theta\quad\Rightarrow\quad dx \:=\:\sec\theta\tan\theta\,d\theta
    . . and: . \sqrt{x^2+1} \:=\:\sec\theta

    Substitute: . \int \tan^3\!\theta \cdot \sec\theta\,(\sec^2\!\theta\,d\theta) \;=\;\int\sec^3\!\theta\tan^3\!\theta\,d\theta

    . . = \;\int\sec^2\!\theta\tan^2\!\theta\,(\sec\theta\ta  n\theta\,d\theta) \;=\;\int\sec^2\!\theta(\sec^2\!\theta - 1)\,(\sec\theta\tan\theta\,d\theta)

    . . = \;\int(\sec^4\!\theta - \sec^2\!\theta)\,(\sec\theta\tan\theta\,d\theta)

    Then let u \,=\,\sec\theta . . .

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: February 19th 2010, 10:55 AM
  2. Integration problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 22nd 2009, 11:03 AM
  3. problem in integration
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 23rd 2008, 09:59 AM
  4. integration problem no.11
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 22nd 2007, 04:02 AM
  5. integration problem no.10
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 22nd 2007, 02:25 AM

Search Tags


/mathhelpforum @mathhelpforum