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Thread: Integration by Parts

  1. #1
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    Integration by Parts

    Hi guys, need some help on these:

    1. $\displaystyle \int x (lnx) dx$

    2. $\displaystyle \int x(cos5x)dx$
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  2. #2
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    Quote Originally Posted by c_323_h
    Hi guys, need some help on these:

    1. $\displaystyle \int x (lnx) dx$
    Let,
    $\displaystyle u=\ln x$ and, $\displaystyle v'=x$
    Thus,
    $\displaystyle u'=1/x$ and, $\displaystyle v=(1/2)x^2$
    Thus,
    $\displaystyle \frac{1}{2}x^2\ln x-\int \frac{1}{2} xdx$
    Thus,
    $\displaystyle \frac{1}{2}x^2\ln x-\frac{1}{4}x^2+C$
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  3. #3
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    Quote Originally Posted by c_323_h

    2. $\displaystyle \int x(cos5x)dx$
    Let,
    $\displaystyle u=x$ and $\displaystyle v'=\cos 5x$
    Thus,
    $\displaystyle u'=1$ and $\displaystyle v=\frac{1}{5}\sin 5x$
    Thus,
    $\displaystyle \frac{1}{5}x\sin 5x-\int \frac{1}{5}\sin 5xdx$
    Thus,
    $\displaystyle \frac{1}{5}x\sin 5x+\frac{1}{25}\cos 5x+C$

    (I am the quickest integrator in the west!)
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  4. #4
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    Quote Originally Posted by ThePerfectHacker
    Let,

    $\displaystyle ...-\int \frac{1}{2} xdx$
    why isn't it $\displaystyle \frac{1}{2}x^2dx$.
    Where did the $\displaystyle x^2$ dissappear to?
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  5. #5
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    Quote Originally Posted by c_323_h
    why isn't it $\displaystyle \frac{1}{2}x^2dx$.
    Where did the $\displaystyle x^2$ dissappear to?
    $\displaystyle \int x^n=\frac{x^{n+1}}{n+1}, n\not =-1$
    Thus,
    $\displaystyle \int xdx=\frac{1}{2}x^2$ but then you need to multiply by $\displaystyle \frac{1}{2}$ thus,
    $\displaystyle \frac{1}{2}\cdot \frac{1}{2}x^2=\frac{1}{4}x^2$
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