Results 1 to 10 of 10

Math Help - need help with integration by parts

  1. #1
    Junior Member
    Joined
    Mar 2007
    Posts
    27

    need help with integration by parts

    Evaluate the following:

    • \int\frac{(ln t)^2}{t}\, dt
    • \int\ln(x^2 + 1)\, dx
    • \int\sin t\, ln(cos t)\, dt
    • \int\frac{cot^-1\sqrt{z}} {\sqrt{z}}\,dz
    • \int\sqrt{x}\ ln x\,dx
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member polymerase's Avatar
    Joined
    May 2007
    From
    Sydney
    Posts
    267
    u=(ln t)^2
    dv=\dfrac{1}{t}
    du=\dfrac{2(ln t)}{t}
    v=ln |t|

    \int\frac{(ln t)^2}{t}\, dt = ln|t|(ln t)^2-\int\frac{2(ln t)^2}{t}\, dt
    3\int\frac{(ln t)^2}{t}\, dt = {ln|t|(ln t)^2}
    \int\frac{(ln t)^2}{t}\, dt = \dfrac{(ln x)^3}{3}

    get the idea?
    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
    12
    Quote Originally Posted by cazimi View Post
    Evaluate the following:
    \int\ln(x^2 + 1)\, dx
    Define f(x)=x\ln(x^2+1)\implies f'(x)=\ln(x^2+1)+\frac{2x^2}{x^2+1}.

    Now \frac{{2x^2 }}<br />
{{x^2 + 1}} = 2\left( {1 - \frac{1}<br />
{{x^2 + 1}}} \right).

    Integrate f'(x),

    f(x)+k=\int\ln(x^2+1)\,dx+2(x-\arctan x).

    And finally \int\ln(x^2+1)\,dx=x\ln(x^2+1)-2(x-\arctan x)+k.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member polymerase's Avatar
    Joined
    May 2007
    From
    Sydney
    Posts
    267
    Let u=ln(x^2+1) and v = x

    Thus,
    du=\dfrac{2x}{x^2+1}dx and dv = dx

    \int\ln(x^2 + 1)\, dx = x\:ln(x^2+1)-\int\dfrac{2x^2}{x^2+1}\, dx
    \int\ln(x^2 + 1)\, dx = x\:ln(x^2+1)-2\int\dfrac{x^2+1-1}{x^2+1}\, dx
    \int\ln(x^2 + 1)\, dx = x\:ln(x^2+1)-2\int1-\dfrac{1}{1+x^2}\, dx
    \int\ln(x^2 + 1)\, dx = x\:ln(x^2+1)-2(x-arctan\:x)
    Therefore \int\ln(x^2 + 1)\, dx = x\:ln(x^2+1)-2x+arctan\:x+C
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    12
    The first one does not require integration by parts, just define u=\ln t and the rest follows.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    12
    Quote Originally Posted by cazimi View Post
    \int\sqrt{x}\ ln x\,dx
    Consider the function f(x)=x^{3/2}\ln x\implies f'(x)=\frac32\sqrt x\ln x+\sqrt x.

    Now apply some of make-up,

    \frac23f'(x)=\sqrt x\ln x+\frac23\sqrt x.

    Integrate, \frac23f(x)+k=\int\sqrt x\ln x\,dx+\frac49x^{3/2}.

    Back substitute

    \int\sqrt x\ln x\,dx=\frac23x^{3/2}\ln x-\frac49x^{3/2}+k.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member polymerase's Avatar
    Joined
    May 2007
    From
    Sydney
    Posts
    267
    Let u=ln(cos\:x) and dv=sin\:dx
    Thus, du=\dfrac{sin\:x}{cos\:x}dx and v=-cos\:x

     \int\sin x\, ln(cos x)\, dx = -cos\:xln(cos\:x)-\int\dfrac{-sin\:x\:cos\:x}{cos\:x}dx
    \int\sin x\, ln(cos x)\, dx = -cos\:xln(cos\:x)+\int\sin\:x\:dx

    Therefore \int\sin x\, ln(cos x)\, dx = -cos\:x[ln(cos\:x)+1] + C
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Mar 2007
    Posts
    27
    Thank you!
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Dec 2007
    Posts
    5

    integration by parts


    find the following genreal anti-derivatives using integration by parts formula


    \int\frac{(x+ 2}{e^3x}\, dx


    Follow Math Help Forum on Facebook and Google+

  10. #10
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,821
    Thanks
    317
    Awards
    1
    Quote Originally Posted by Sundevils View Post
    find the following genreal anti-derivatives using integration by parts formula


    \int\frac{(x+ 2}{e^3x}\, dx


    Please do not double post. See rule #1 here.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: January 11th 2012, 02:30 PM
  2. Replies: 8
    Last Post: September 2nd 2010, 12:27 PM
  3. Replies: 0
    Last Post: April 23rd 2010, 03:01 PM
  4. Integration by Parts!
    Posted in the Calculus Forum
    Replies: 7
    Last Post: January 22nd 2010, 03:19 AM
  5. Replies: 1
    Last Post: February 17th 2009, 06:55 AM

Search Tags


/mathhelpforum @mathhelpforum