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Math Help - Help with integrals

  1. #1
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    Help with integrals

    How do I integrate:

    ∫ 2x/(x-4) dx

    and

    ∫ (x+3)sqrt(x-2) dx



    Thanks for help!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Kenny12345 View Post
    How do I integrate:

    ∫ 2x/(x-4) dx

    and

    ∫ (x+3)sqrt(x-2) dx



    Thanks for help!
    For the first one, make the substitution u=x-4\implies x=u+4 Thus, \,du=\,dx

    The integral now becomes \int\frac{2\left(u+4\right)}{u}\,du=\int\left(1+\f  rac{8}{u}\right)\,du. Can you take it from here?

    For the second one, left u=x-2\implies x=u+2. Thus, \,du=\,dx

    The integral now becomes \int\left[\left[\left(u+2\right)+3\right]\sqrt{u}\right]\,du=\int\left(u^{\frac{3}{2}}+5u^{\frac{1}{2}}\ri  ght)\,du. Can you take it from here?

    Does this make sense?
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    For the first one, make the substitution u=x-4\implies x=u+4 Thus, \,du=\,dx

    The integral now becomes \int\frac{2\left(u+4\right)}{u}\,du=\int\left(1+\f  rac{8}{u}\right)\,du. Can you take it from here?

    For the second one, left u=x-2\implies x=u+2. Thus, \,du=\,dx

    The integral now becomes \int\left[\left[\left(u+2\right)+3\right]\sqrt{u}\right]\,du=\int\left(u^{\frac{3}{2}}+5u^{\frac{1}{2}}\ri  ght)\,du. Can you take it from here?

    Does this make sense?
    For the second question, can I also use integration by parts as well?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Kenny12345 View Post
    For the second question, can I also use integration by parts as well?
    You could...but it would over-complicate things. A substitution would be easier (and quicker).
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    Quote Originally Posted by Chris L T521 View Post
    You could...but it would over-complicate things. A substitution would be easier (and quicker).
    Alright thanks, I have another question:

    ∫ x^7sin(3x^4) dx

    I'm supposed to use integration by parts for this one, but I'm not sure what I should take for u and v first.
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Kenny12345 View Post
    Alright thanks, I have another question:

    ∫ x^7sin(3x^4) dx

    I'm supposed to use integration by parts for this one, but I'm not sure what I should take for u and v first.
    I would first tweek the integral before I apply integration by parts:

    let z=3x^4\implies x^4=\tfrac{1}{3}z. Thus, \,dz=12x^3\,dx\implies x^3\,dx=\tfrac{1}{12}\,dz

    Now, the integral becomes \frac{1}{12}\int x^4\sin\left(z\right)\,dz=\frac{1}{36}\int z\sin\left(z\right)\,dz

    Now integration by parts will be a piece of cake from here.

    Let u=z and \,dv=\sin\left(z\right)

    Why don't you take it from here and find \,du and v.



    PS: In the future, if you have new questions, please ask them in a new thread.
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  7. #7
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    Quote Originally Posted by Chris L T521 View Post
    I would first tweek the integral before I apply integration by parts:

    let z=3x^4\implies x^4=\tfrac{1}{3}z. Thus, \,dz=12x^3\,dx\implies x^3\,dx=\tfrac{1}{12}\,dz

    Now, the integral becomes \frac{1}{12}\int x^4\sin\left(z\right)\,dz=\frac{1}{36}\int z\sin\left(z\right)\,dz

    Now integration by parts will be a piece of cake from here.

    Let u=z and \,dv=\sin\left(z\right)

    Why don't you take it from here and find \,du and v.



    PS: In the future, if you have new questions, please ask them in a new thread.
    Yea sorry, I'm a newbie in these forums and I thought it would save post space if I just reply similar questions in a single post instead of making a new thread.
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  8. #8
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Kenny12345 View Post
    Alright thanks, I have another question:

    ∫ x^7sin(3x^4) dx

    I'm supposed to use integration by parts for this one, but I'm not sure what I should take for u and v first.
    Here's something that may be helpful in the future:

    To determine what u should be, you can use the anagram LIPET. Go down the list (in this particular order) and see what you have in the integral [i.e. if you don't have logs, do you have inverse trig? etc]:

    Logarithms (Natural) --------------Highest priority
    Inverse Trigonometric Functions
    Polynomial Expressions
    Exponential Functions
    Trigonometric Functions -----------Lowest Priority
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