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Math Help - [SOLVED] Tech. of Integration

  1. #1
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    [SOLVED] Tech. of Integration

    xe^(2x)/ (2x+1)^2
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  2. #2
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by swensonm View Post
    xe^(2x)/ (2x+1)^2
    \int{\frac{e^{2x}\times x}{(2x+1)^2}}
    -----------------------------
    This is a simple question if you remember that
    whenever you get an integral with e^x

    Try this

    \int{e^{x} (f(x) +f'(x) )} = e^x f(x) + C

    Try proving it its fun
    -----------------------
    Here let 2x =t

    2dx =dt
    dx =dt/2

    (2x+1) = t+ 1


     <br />
\int{\frac{e^{t}\times t dt }{4(t+1)^2}}<br />


    All you need to do now is See the form , try it before looking down



    ----------------------------------------
    Here

     <br />
\int{e^{t}~\frac{( t dt) }{4(t+1)^2}}<br />

     <br />
\int{e^{t}~\frac{(t+1-1) dt }{4(t+1)^2}}<br />

     <br />
\int{\frac{e^{t}}{4}(\frac{ t+1}{(t+1)^2}-\frac{ 1 }{(t+1)^2}})dt<br />


    Can you see something
    -------------------------------------------------------------

     <br />
\frac{d}{dt} \{\frac{ 1 dt }{(t+1)}\} = \frac{ -1}{(t+1)^2}<br />

    So ultimately
    integration has become

     <br />
\int{\frac{e^{t}}{4}(f'(t)+f(t))}<br />

    Answer will be

    e^{t} f(t)/4 = \frac{ e^{t} \{ \frac{ 1 }{ (t+1)} \} }{4}

    Put the value of t = 2x

    e^{2x} f(2x)/4 = \{ \frac{ e^{2x} } { 4(2x+1)} \}
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ADARSH View Post
    \int{\frac{e^{2x}\times x}{(2x+1)^2}}
    -----------------------------
    This is a simple question if you remember that
    whenever you get an integral with e^x

    Try this

    \int{e^{x} (f(x) +f'(x) )} = e^x f(x) + C

    ...
    Nice method. You would probably "see" it easier if you let t = 2x + 1 though. no need for that "+1 - 1" maneuver then
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