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Math Help - integration question

  1. #1
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    integration question

     \int_1^{7} f(x) dx = 19 and \int_5^{7} f(x)dx = 12

    then find
     \int_1^{5} 3f(x) dx

     \int_1^{7} f(J) dJ

    how would i work this out? cheers
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  2. #2
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    Any constant can be moved inside and outside of an integral freely. So:

    \int_1^{5} 3f(x) dx = 3 \int_{1}^{5}f(x)dx

    Now if we simply use the definition of integral bounds, we can construct these bounds by combining others.

    \int_{1}^{5}f(x)dx=\int_{1}^{7}f(x)dx-\int_{5}^{7}f(x)dx=19-12=7

    Multiply by 3 and I get an answer of 21.
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  3. #3
    Junior Member toraj58's Avatar
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    i have depicted this problem for you visually to understand it.
    please see the attachement image.
    Attached Thumbnails Attached Thumbnails integration question-integral.jpg  
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  4. #4
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    Quote Originally Posted by Jameson View Post
    Any constant can be moved inside and outside of an integral freely. So:

    \int_1^{5} 3f(x) dx = 3 \int_{1}^{5}f(x)dx

    Now if we simply use the definition of integral bounds, we can construct these bounds by combining others.

    \int_{1}^{5}f(x)dx=\int_{1}^{7}f(x)dx-\int_{5}^{7}f(x)dx=19-12=7

    Multiply by 3 and I get an answer of 21.
    ahh okay, do you do the same thing for part (b)? \int_1^7 f(J) dJ ?

    EDIT: Sorry i understand now \int_1^7 f(J) dJ = 19
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  5. #5
    Junior Member toraj58's Avatar
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    yes it is possible to change x with another variable like j and the intgral remain the same.
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