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Math Help - Integration using substitution

  1. #1
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    Integration using substitution

    The following definite integral can be evaluated to give the answer A - B where A and B are found from substituting the limits of integration.

    Integration using substitution-bafadc17fcd7f5f278d51680c123321.png

    What are A and B?

    Having a difficult time showing how to integrate this to get the answer
    3ln(7x^2 + 5x + 3). The denominator can be represented by u and du would be 14x + 5. With this, I divide and multiply it all by 14 I believe to get the 3 thats in the answer. This will give me the integral of 3/u. According to integration rules, 1/u is equal to the ln of u. Thus i get 3ln(u) and sub back in the original values. Then you evaluate at 3 and 0 to get A and B. Is this correct? If it doesnt make sense then let me know I can try to clarify. Thankss!
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  2. #2
    Super Member TheChaz's Avatar
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    I think this is what they are getting at... we have

    \int_{0}^{3}\frac{42x^2 + 15}{7x^2 + 5x + 3}dx

    And decide to let u = 7x^2 + 5x + 3
    Then du = (14x + 5)dx so our integral becomes...

    3*\int_{?}^{?}\frac{du}{u}

    Notice that I have left the limits of integration out. They are no longer 0 and 3. One way to prevent errors would just be to always explicitly write
    \int_{x = 0}^{x = 3}\frac{42x^2 + 15}{7x^2 + 5x + 3}dx

    But that's tedious.

    So we will see that, when x = 0, u = 3 (from our choice of u sub above).
    Likewise, when x = 3, u = ...

    THOSE are the "A" and "B" to which they refer.
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  3. #3
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    Thanks for the help. I think I got it.
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  4. #4
    Super Member TheChaz's Avatar
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    You are welcome.
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