# Integration by Parts & Change of Variables Proof

In summary, The conversation is about the proofs of the Integration by Parts and Change of Variables formulas as given in a book. The speaker has uploaded their own rewrite of the proofs but is unsure if it is correct. They are seeking help in identifying any errors and understanding why they were made. Another person suggests using the product rule for differentiation or the fundamental theorem of calculus instead. The reason for using Integration by Parts is seen as a challenge.
I'm just curious about the proofs of Integration by Parts & the Change of Variables formula
as given in this book on page 357. I think there are a lot of typo's so I've uploaded my
rewrite of them but I am unsure of how correct my rewrites are. If someone could point
out the errors & why I made them I'd really appreciate it as I don't see how I messed up
but still feel I did, for example I'm not sure whether the $$g_{i-1}$$ is a typo in
the last lines of the IBP proof, when the author takes the norm of the partition to zero.
I can't see how it's included according to my rewrite

For integration by parts, why not just integrate the product rule for differentiation?

The challenge I suppose. So no errors?

I don't see why you just done use the fundamental theorem of calculus. That's what we did in our analysis course.

## 1. What is the concept of integration by parts?

Integration by parts is a technique used to solve integrals that involve products of functions. It is based on the product rule of differentiation, and involves breaking down an integral into two parts and using a specific formula to solve it.

## 2. How do you use the integration by parts formula?

The integration by parts formula is ∫udv = uv - ∫vdu, where u and v are two functions that make up the integrand. To use this formula, you must choose u and dv in a way that makes the integral easier to solve. Then, you can plug in the values for u, dv, v, and du to evaluate the integral.

## 3. What is the change of variables method in integration?

The change of variables method, also known as u-substitution, is a technique used to solve integrals by substituting a variable in the integrand with a new variable. This new variable is chosen in a way that simplifies the integral and makes it easier to solve.

## 4. How do you prove the integration by parts and change of variables formulas?

The proofs for both integration by parts and change of variables involve using the fundamental theorem of calculus and the chain rule of differentiation. By applying these rules and manipulating the integrals, the formulas can be derived and proven to be true.

## 5. When is it best to use integration by parts versus the change of variables method?

Integration by parts is typically used when the integrand can be split into two parts, while the change of variables method is useful when the integrand contains a complicated function. Both methods can also be used together to solve more complex integrals. The best approach will depend on the specific integral and the functions involved.

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