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    Does the integral of v(du/dx) w.r.t x equal the integral of v w.r.t u? (I think I saw this rule somewhere, not sure where :/ )
    Or is this too general haha ?
    Thanks for any response
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    if this is what you meant: integration by parts: \int u\frac{dv}{dx}dx = uv - \int v\frac{du}{dx}dx

    Write dx instead of w.r.t. x.

    Your rule does work, because the dx and \frac{d}{dx} cancel. It's the opposite of integration by substitution.
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    W/o considering differentiability of v w.r.t u.

    If u is a function of x, then \dfrac{du}{dx} = u'(x) \, \, \, \Rightarrow du = u'(x) dx as u = u(x)

    \displaystyle \int vu'(x) \, dx \Rightarrow \int vdu
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    (Original post by Phichi)
    W/o considering differentiability of v w.r.t u.

    If u is a function of x, then \dfrac{du}{dx} = u'(x) \, \, \, \Rightarrow du = u'(x) dx as u = u(x)

    \displaystyle \int vu'(x) \, dx \Rightarrow \int vdu
    Thanks- this is what I hoped!
    (Original post by morgan8002)
    if this is what you meant: integration by parts: \int u\frac{dv}{dx}dx = uv - \int v\frac{du}{dx}dx

    Write dx instead of w.r.t. x.

    Your rule does work, because the dx and \frac{d}{dx} cancel. It's the opposite of integration by substitution.
    Why write dx instead of w.r.t. x - they mean the same don't they?
    I suppose when writing the integral one would obviously write dx lol.
    Thanks. Yh I was curious about IBP and asked out substitution formula I suppose. d/dx cancel? dx is not a variable like in algebra so I suppose you mean during the working out or something
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    (Original post by MathMeister)
    Why write dx instead of w.r.t. x - they mean the same don't they?
    I suppose when writing the integral one would obviously write dx lol.
    Thanks. Yh I was curious about IBP and asked out substitution formula I suppose. d/dx cancel? dx is not a variable like in algebra so I suppose you mean during the working out or something
    Make it look like I'm blowing my own trumpet in my quote
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    (Original post by Phichi)
    Make it look like I'm blowing my own trumpet in my quote
    lol
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    (Original post by MathMeister)
    Thanks- this is what I hoped!

    Why write dx instead of w.r.t. x - they mean the same don't they?
    I suppose when writing the integral one would obviously write dx lol.
    Thanks. Yh I was curious about IBP and asked out substitution formula I suppose. d/dx cancel? dx is not a variable like in algebra so I suppose you mean during the working out or something
    dx is more concise and is easier to work with.

    The differentials aren't fractions, but can sometimes be manipulated like fractions. Phichi had more detailed explanation, so I won't go into the details.
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    (Original post by MathMeister)
    d/dx cancel? dx is not a variable like in algebra so I suppose you mean during the working out or something
    \dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac{du}{dx}

    dx is just as \delta x \rightarrow 0, it can be treated as that still in cases. It's a touchy subject, dx itself is pretty meaningless, but \dfrac{dy}{dx} has a condition built in, and has to be treated differently.

    (Original post by morgan8002)
    dx is more concise and is easier to work with.
    w.r.t is usually the appropriate syntax in text when notation isn't used.
 
 
 
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