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And in general when working with integration techniques, does an expression like du have any mathematical meaning by itself or are those substitutions just used to make integration easier?

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- Thread starter Polymath89
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In summary, integration by parts is a technique that allows for the easier solving of a particular integral term. The substitution of u = 3x2 makes the integral simpler to work with. and the lack of mathematical meaning is due to the fact that most people learn calculus after it has been applied.f

- #1

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And in general when working with integration techniques, does an expression like du have any mathematical meaning by itself or are those substitutions just used to make integration easier?

- #2

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Integration by parts is essentially the integral counterpart of the product rule in differentiation.I have some problems understanding the intuition behind the integration by parts technique. I don't quite see why you solve for [itex]\int u(x)v\prime (x)[/itex], instead of one of the other parts, what makes it easier to solve for that particular term?

If f(x) = u(x) * v(x), then f'(x) = u(x) * v'(x) + u'(x) * v(x)

If we integrate the equation above, we get

∫f'(x)dx = ∫(u(x) * v'(x) + u'(x) * v(x))dx

= ∫u(x) * v'(x) dx + ∫u'(x) * v(x)dx

Simplifying a bit, we get

f(x) = ∫u(x) * v'(x) dx + ∫u'(x) * v(x)dx

or

∫u(x) * v'(x) dx = f(x) - ∫u'(x) * v(x)dx = u(x)*v(x) - ∫u'(x) * v(x)dx

Notice that I solved for ∫u(x) * v'(x) dx. I could just as easily have solved for the other integral. So in answer to your question, it doesn't make any difference. The only criterion is that ∫u'(x) * v(x)dx should be easier to work with than the other integral.

Yes, it has meaning. When you use a substitution, say u = 3xAnd in general when working with integration techniques, does an expression like du have any mathematical meaning by itself or are those substitutions just used to make integration easier?

- #3

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As for your comment about the mathematical meaning of du, its understandable to have the feeling that it has no mathematical meaning because most people don't see the meaning until they are done learning calculus and are instead applying it.

It is a differential element of u. What does that mean? I think of it as "a piece of u so small that it can barely be said to exist at all."

Look at it this way - the derivative notation, dy/dx, is a slope. The reason why we are saying "differential element of y over a differential element of x" is because even though the rate of change of a function may not be constant, the differential elements are

- #4

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Thats was pretty helpful, thanks a lot guys.

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