- #1

Sang Ho Lee

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This is the example he gave"

Is there a name for this technique that substitutes d(___) instead of dx?

Thank you,

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In summary, the conversation revolved around a new technique for solving Integration By Parts without explicitly stating dv. This technique substitutes d(___) instead of dx, but it was clarified that this is just another way of writing dv. The speaker prefers to use the "reverse product rule" and chooses variables based on their ease of differentiation and integration. They also provided a proof for integration by parts as a way to understand the process better.

- #1

Sang Ho Lee

- 4

- 0

This is the example he gave"

Is there a name for this technique that substitutes d(___) instead of dx?

Thank you,

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- #2

Mark44

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It doesn't substitute d(___) instead of dx. What you're calling d(___) is just dv.Sang Ho Lee said:

This is the example he gave"

Is there a name for this technique that substitutes d(___) instead of dx?

Thank you,

This is nothing more than integration by parts, but without stating dv explicitly. Here ##u = \ln x## and ##dv = xdx##. From the latter, we get ##v = \frac{x^2}2##.

- #3

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##\int f(x)g(x) dx = f(x) \int g(x)dx - \int f'(x) (\int g(x)dx)dx## (*)

I don't see why you need any other variables. You pick ##f## because it's easy to differentiate and/or ##g## because you know how to integrate it already. Your one I would do:

##\int (ln(x)) x dx = (ln(x)) \frac{x^2}{2} - \int (\frac{1}{x})(\frac{x^2}{2})dx = \frac{x^2ln(x)}{2} - \int \frac{x}{2} dx = \frac{x^2ln(x)}{2} - \frac{x^2}{4} + C##

I knew how to differentiate ##ln(x)## and how to integrate ##x##.

PS To prove (*), I would imagine that you already know the integral of ##g## is ##h##, which means ##h'(x) = g(x)##. So:

## \int f(x)g(x) + f'(x)h(x)dx = \int f(x)h'(x) + f'(x)h(x)dx = \int \frac{d}{dx} (f(x)h(x))dx = f(x)h(x) + C##

Hence:

## \int f(x)g(x)dx = f(x)h(x) -\int f'(x)h(x)dx+ C##

Where ##\int g(x)dx = h(x) + C##

And that seems to me to take some of the "magic" out of integration by parts.

Integration by Parts is a technique used to evaluate integrals that can be written as the product of two functions. It involves rewriting the integral in a specific form and applying a formula to solve it. In this method, the functions are chosen based on their derivatives and integrals, rather than assigning them as u and v.

Using Integration by Parts without u, v can be helpful when the given functions do not easily fit into the standard u-substitution or when the integrand involves trigonometric functions. It also eliminates the need to choose which function to assign as u and v, making the process simpler and less prone to error.

The functions are chosen based on their derivatives and integrals. One function is chosen based on its derivative, while the other is chosen based on its integral. This method is often referred to as the ILATE rule, where I stands for inverse trigonometric functions, L stands for logarithmic functions, A stands for algebraic functions, T stands for trigonometric functions, and E stands for exponential functions.

While Integration by Parts can be used to evaluate a wide range of integrals, there are certain integrals where this method may not be the most efficient or effective. For example, when the integral involves a product of two polynomials, it is generally easier to use algebraic manipulation to solve it rather than Integration by Parts.

One limitation of this method is that it can sometimes lead to a more complicated integral than the original one. It also requires a good understanding of the properties of different functions and their derivatives and integrals. Additionally, it may not always be the most efficient method for solving integrals, depending on the complexity of the functions involved.

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