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Math Help - simple rational integration

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    simple rational integration

    Hello, I have completely forgotten the procedure for integrating fractions such as the following. Can someone please show me the procedure. Thanks kindly for any help.

    1/x
    answer = (x^2)/4

    and

    sin3x / 3
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    Re: simple rational integration

    Quote Originally Posted by fran1942 View Post
    Hello, I have completely forgotten the procedure for integrating fractions such as the following. Can someone please show me the procedure. Thanks kindly for any help.

    1/x
    answer = (x^2)/4

    and

    sin3x / 3
    The answer to your first one is wrong. \displaystyle \begin{align*} \frac{d}{dx}\left(\ln{|x|}\right) = \frac{1}{x} \end{align*}. What does that tell you about \displaystyle \begin{align*} \int{\frac{1}{x}\,dx} \end{align*}?

    For \displaystyle \begin{align*} \int{\frac{\sin{(3x)}}{3}\,dx} = \frac{1}{9}\int{3\sin{(3x)}\,dx} \end{align*}, use a substitution \displaystyle \begin{align*} u = 3x \implies du = 3\,dx \end{align*}.
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    Re: simple rational integration

    thanks, I understand sin3x/3,. Is the integral of 1/x simply ln(x) ?

    If so how would that apply to integrating 'x/3' ?

    Thanks for your help.
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    Re: simple rational integration

    Quote Originally Posted by fran1942 View Post
    thanks, I understand sin3x/3, but I am still confused over how to integrate 1/x.

    If you could show me the steps involved in that and also how to integrate 'x/3', then I will get it.

    Thanks for your help.
    Do you understand the integration is the opposite of differentiation?

    Also, x/3 = (1/3)x. Surely you can integrate that...
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    Re: simple rational integration

    Quote Originally Posted by fran1942 View Post
    thanks, I understand sin3x/3,. Is the integral of 1/x simply ln(x) ?

    If so how would that apply to integrating 'x/3' ?

    Thanks for your help.
    It appears that you have edited your post. No, the integral of 1/x is NOT ln(x). What happens when x < 0?

    What is the function that I wrote down which differentiates to give 1/x?
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