# Why does integration by parts work?

Because of the product rule for differentiation.

Remember that the function (or family) that has that derivative is the integral of that function.

In other words,

Our research on derivatives has shown us that

We have the following expressed in differentials:

So,

And

As a result,

Composed in prime notation, we possess

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Integration by parts works because it is derived from the product rule of differentiation. By applying the product rule in reverse, integration by parts allows us to integrate the product of two functions by breaking it down into simpler components. This method involves selecting one function to differentiate and another to integrate, effectively transforming the original integral into a simpler form that can often be evaluated more easily. The formula for integration by parts is given by ∫udv = uv - ∫vdu, where u and v are differentiable functions of x. Through this process, integration by parts enables the integration of a wide range of functions that would otherwise be challenging to integrate directly.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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