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    Hello, tis me again - with some more stinky integration !! I'm getting a tad confused my first problemo is what method to use to work out:

    1) ∫[ (x+1)²/(x²+1) ] dx
    I had a go at doing it by substitution and but that really wasnt looking so pretty... so yeah any suggestions about the method I could use...

    2) ∫cosxsinx
    I know this is easy, I can see that it is, but brain has gone into overdrive. I dont know how to set out the answer - cos you can kinda just see that the 2 parts are related... do I have to actually write that?

    Thanks for your help
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    (Original post by franks)
    Hello, tis me again - with some more stinky integration !! I'm getting a tad confused my first problemo is what method to use to work out:

    1) ∫[ (x+1)²/(x²+1) ] dx
    I had a go at doing it by substitution and but that really wasnt looking so pretty... so yeah any suggestions about the method I could use...

    2) ∫cosxsinx
    I know this is easy, I can see that it is, but brain has gone into overdrive. I dont know how to set out the answer - cos you can kinda just see that the 2 parts are related... do I have to actually write that?

    Thanks for your help
    1) ∫cosxsinx dx
    = ∫½sin2x dx
    = -(1/4)cos2x + c

    2) ∫(x+1)²/(x²+1) dx
    = ∫(x²+1+2x)/(x²+1) dx
    = ∫(x²+1)/(x²+1) + 2x/(x²+1) dx
    = ∫1 + 2x/(x²+1) dx
    = x + ln(x²+1) + c
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    1) ∫[ (x+1)²/(x²+1) ] dx
    ∫[(x²+2x+1)/(x²+1)] dx = ∫[1+2x/(x²+1)] dx NOTE d/dx(x²+1) = 2x
    =x + ln(x²+1) + C

    2) ∫cosxsinx
    ∫cosxsinx dx = ∫½sin2x dx
    = -¼cos2x + C

    Hope this helps
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    but how did you work it out ??? Especially ∫(x+1)²/(x²+1) dx ?! You just seem to have arrived at the answer - I have NO idea how to get there

    --------------

    oh dear, I'm so confused
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    (Original post by franks)
    but how did you work it out ??? Especially ∫(x+1)²/(x²+1) dx ?! You just seem to have arrived at the answer - I have NO idea how to get there

    --------------

    oh dear, I'm so confused
    see my post.
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    (Original post by franks)
    but how did you work it out ??? Especially ∫(x+1)²/(x²+1) dx ?! You just seem to have arrived at the answer - I have NO idea how to get there

    --------------

    oh dear, I'm so confused
    Expand the brackets then long division.
    then manipulate the standard result of ∫1/x dx = ln(X) + C
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    I dont know how to integrate 2x/(x²+1) ... thats part of the problem - I thought you couldnt do it with ln|x| ...

    --------------

    oh no, i do i do !

    --------------

    its the one with the special rule - cos the top's the differential of the bottom whupeeee thanks for helping
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    You use the fact that ∫f'(x)/f(x)dx= ln[f(x)] + c (this comes from differentiating ln[f(x)]). In the case of 2x/(x²+1), if f(x)=(x²+1) then f'(x)=2x. Therefore ∫2x/(x²+1)dx = ln[x²+1] + c.
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    stuck again
    I have absolutely no idea how to do this one - I literally have pages of working, Ive tried substitution, recognition, everything!

    ∫[(sec²x)/(1+tanx)³]

    I also tried splitting up the bottom line and then swapping tan into sec² but that just ended up with an equation about 4 lines long
    I hate this question so much!!
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    You can do it by inspection. Notice:

    ∫[(sec²x)/(1+tanx)³]=∫(sec²x)(1+tanx)-3
    And that sec²x is the differential of (1+tanx).

    Then think about what needs to be differentiated to get (sec²x)/(1+tanx)³.

    Spoiler:
    Show
    ∫[(sec²x)/(1+tanx)³]dx= -0.5(1+tanx)-2+C
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    thank youuu all sorted !
 
 
 
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