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Math Help - Integration

  1. #1
    Len
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    Integration

    Alright so I'm a little confused with basic integration.

    \int\frac{dx}{x}=ln|x|+c
    But what if theres a constant infront of dx.
    For example
    \int\frac{7dx}{x}=\int\frac{1}{x}*7dx=?

    What changes then?
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  2. #2
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    As 7 is a constant, it is not affected by the integration, so you can do this:

    <br />
\int\frac{7dx}{x}

    = 7\int\frac{dx}{x}

    = 7ln|x| + c
    = ln |x^7| + c<br />

    or if you like to think of it this way:

    7\int\frac{7dx}{x}(\frac{1}{7})
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  3. #3
    Len
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    Quote Originally Posted by Gusbob View Post
    As 7 is a constant, it is not affected by the integration, so you can do this:

    <br />
\int\frac{7dx}{x}

    = 7\int\frac{dx}{x}

    = 7ln|x| + c
    = ln |x^7| + c<br />

    or if you like to think of it this way:

    7\int\frac{7dx}{x}(\frac{1}{7})
    Thanks, another quick question,
    \int\frac{7dx}{4x+2}=?=7ln|4x+2|
    Or do I have to multiply it by the derivative of 4x+2?
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  4. #4
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    <br />
\int\frac{7dx}{4x+2}<br />

    You multiply by the derivative, but don't forget to divide it.

    For this one, I'll take the 7 out again and use the substitution u = 4x + 2. du = 4 dx.

    You need a 4 on top, so multiply by \frac{4}{4}.

    <br />
7\int\frac{4dx}{4x+2}(\frac{1}{4})<br />

    Since 1/4 is a constant, I took it out like I did with the 7. (Note: only constants that are multiplied or divided can be taken out, addition or subtraction of constants are integrated.)

    \frac{7}{4} \int\frac{du}{u} (since du = 4dx; u = 4x+2)

    = \frac{7}{4} ln|4x+2| +c
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  5. #5
    Len
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    Quote Originally Posted by Gusbob View Post
    <br />
\int\frac{7dx}{4x+2}<br />

    You multiply by the derivative, but don't forget to divide it.

    For this one, I'll take the 7 out again and use the substitution u = 4x + 2. du = 4 dx.

    You need a 4 on top, so multiply by \frac{4}{4}.

    <br />
7\int\frac{4dx}{4x+2}(\frac{1}{4})<br />

    Since 1/4 is a constant, I took it out like I did with the 7.

    \frac{7}{4} \int\frac{du}{u} (since du = 4dx; u = 4x+2)

    = \frac{7}{4} ln|4x+2| +c
    Thanks a ton, things are a lot clearer now!
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    I dont know it this helps

    but. \int\frac{1}{ax+b}dx= \frac{1}{a}\ln(ax+b)\ using the same technique by isolating the derivative of the quantity
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