How to integrate a function with a square root in it

  1. rock.freak667

    rock.freak667 6,230
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    1. The problem statement, all variables and given/known data

    find [tex] \int x^{\frac{3}{2}}\sqrt{1+x} dx [/tex]

    2. Relevant equations



    3. The attempt at a solution

    Now i tried all the methods i know of which include integration by parts and substitution,the integration by parts didn't work too well, so substitution I tried.
    let [tex]u=tan^2x[/tex] and [tex]du = 2tanxsec^2x[/tex]

    eventually giving me
    [tex]\int tan^4xsec^3x dx[/tex] which i can't do

    and if I used [tex]u^2=x+1[/tex] I eventually get

    [tex]2\int u^3(u^2-1)^\frac{3}{2} du[/tex] which i also can't do

    is there any useful substitution i can do?
     
  2. jcsd
  3. rock.freak667

    rock.freak667 6,230
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  4. hm..it seems they did integration by parts with u=x^n and dv=square root but I can't follow it from there...

    But why can't you do the first integral? you can just change tan, sec to sin and cos's and then do a substitution which will give you like..3 fractions of u to some power divided by u to some greater power.
     
    Last edited: Sep 3, 2007
  5. rock.freak667

    rock.freak667 6,230
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    because then i will get

    [tex]\int (sec^7\theta -2sec^5\theta + sec^3\theta)d\theta[/tex]
     
  6. No just keep it as:

    [tex]\int\frac{tan^4 x}{sec^3 x} dx[/tex]

    but write tan and sec as sin/cos to give you:

    [tex]\int\frac{sin^4 x}{cos^4 x}\frac{1}{cos^3} dx [/tex]

    and then simplify those fractions into 1, then do a substitution.
     
  7. rock.freak667

    rock.freak667 6,230
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    Wouldn't simplifying it bring it back to an expression with tan and sec in it
     
  8. no what do you get if you simplify [itex]cos^4 x * cos^3 x[/itex]? then you can do the substitution.
     
  9. rock.freak667

    rock.freak667 6,230
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    Well I would get [tex] cos^3\theta-2sin^2\thetacos^2\theta+sin^4\thetacos^3\theta[/tex] by using [tex] cos^2\theta+sin^2\theta=1[/tex]

    or should i get a substitution for [tex]cos^3 x[/tex]
     
  10. if you have [itex]cos^n x * cos^m x[/itex] you simplify it the same as: [itex]u^n*u^m[/itex] which is equal to?
     
  11. rock.freak667

    rock.freak667 6,230
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    [tex]u^n*u^m = u^{n+m}[/tex] thus [tex]cos^3\theta * cos^4\theta = cos^7\theta[/tex]
     
  12. yep so now you have
    [tex]\int\frac{sin^4 x dx}{cos^7 x}[/tex]

    so what substitution would be nice there?
     
  13. rock.freak667

    rock.freak667 6,230
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    well normally I'd say [tex]u=cosx[/tex] but the high powers of the integrand is confusing me a bit
     
  14. Hm..ok just saw a bit of a problem...

    when you do u=cos x the top sin^4 x has to be split to sin x * sin^2 x * sin x which will be a square root * a polynomial.

    also that reduction formula you found might not work since it needs x^(n-1) and if you use it you'll get x^(1/2) and x^(-1/2) so you wont hit x^0 which is needed....
     
  15. rock.freak667

    rock.freak667 6,230
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    so...there really is no way to integrate this?
     
  16. There is an analytical answer, but it is a pretty nasty integral judging by the answer mathematica gives. I can't immediately think of a good trig substitution or good way to go about it. It might end up being that you need to use a mixture of integration by parts and some form of substitution, but if this is a problem that you set up I would make sure that everything leading up to the equation is right.
     
  17. the integrator from the Mathematica website can do it but I have no idea how...

    seems like you're going to get some sort of square root * polynomial and then + some hyperbolic function...
     
  18. rock.freak667

    rock.freak667 6,230
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    Well this is just one question that my Further math teacher gave me..although he doesn't show you how to do them if you don't get it out
     
  19. I got a solution, but damn it was hard. I am going to try to find an easier method.
     
  20. Tried x=(sinh(u))^2 and it works through to the same result as Mathematica.
    After substitution you convert the sinh's, cosh's to exponents using sinh(u)=0.5*(exp(u)-exp(-u)), cosh(u)=0.5*(exp(u)+exp(-u)).
    You get a simple integral but then you have to work to convert the sinh's and cosh's of multiples of u to powers of sinh(u) and cosh(u) but it works out fine.
     
  21. How did you get u=tan^2(x)??

    because x is really [sqrt(x)]^2 having trouble seeng the tan^2(x).
     
    Last edited: Sep 4, 2007
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