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Math Help - Some integration help.

  1. #1
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    Some integration help.

    Hi,
    I have two questions I'd like some help with.
    Firstly what is \int cos^3 2x dx ?
    I tried doing it with trigonometric identities for cos 2x but it became very nasty pretty fast. Any help would be great.


    Also, this has been bugging me all day, and I can't see where I am wrong, though I must be... Why is the integral of \int \tfrac{2}{2x+3} dx  =  ln(2x+3) yet \int \tfrac{1}{x+1.5} dx  =  ln(x+1.5)

    Because \tfrac{2}{2x+3}  =  \tfrac{1}{x+1.5} yet  ln(x+1.5) is not the same as ln(2x+3)

    What am I doing wrong?

    Thanks.
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  2. #2
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    Quote Originally Posted by lord_day View Post
    Hi,
    I have two questions I'd like some help with.
    Firstly what is \int cos^3 2x dx ?
    I tried doing it with trigonometric identities for cos 2x but it became very nasty pretty fast. Any help would be great.

    [snip]
    \cos^3 (2x) = \cos^2 (2x) \, \cos (2x) = [1 - \sin^2 (2x)] \, \cos (2x).

    Now make the substitution u = \sin (2x).
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  3. #3
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    Quote Originally Posted by lord_day View Post
    [snip]
    Also, this has been bugging me all day, and I can't see where I am wrong, though I must be... Why is the integral of \int \tfrac{2}{2x+3} dx = ln(2x+3) yet \int \tfrac{1}{x+1.5} dx = ln(x+1.5)

    Because \tfrac{2}{2x+3} = \tfrac{1}{x+1.5} yet  ln(x+1.5) is not the same as ln(2x+3)

    What am I doing wrong?

    Thanks.
    What you're doing wrong is forgetting about the arbitary constant of integration.

    \int \tfrac{2}{2x+3} dx = \ln |2x+3| + C_1 = \ln |2(x + 1.5)| + C_1 = \ln (2) + \ln |x + 1.5| + C_1 = \ln |x + 1.5| + C_2 where C_2 = \ln (2) + C_1 is just as arbitrary as C_1.

    \int \tfrac{1}{x+1.5} dx = \ln |x+1.5| + C_2.

    What do you notice?

    BY the way, notice the use of the absolute value brackets NOT round brackets. The answers you gave are therefore also wrong for this reason too.
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