Explain to me how integration by parts works in Calculus, with an example
Integration by parts is a technique in calculus used to integrate a product of two functions. It involves rewriting the integral of the product of two functions as an expression involving only one of the functions, and its derivative, and the other function and its antiderivative.

The formula for integration by parts is:

∫u dv = uv − ∫v du

where u and v are two functions, and du/dx and dv/dx are their derivatives with respect to x.

To use this formula, we choose one function to be u and the other to be dv. Then, we differentiate u and integrate dv.

Here is an example:

∫x sin x dx

In this example, we choose u = x and dv = sin x dx.

Then, we have du/dx = 1 and v = -cos x.

Applying the formula, we get:

∫x sin x dx = -x cos x + ∫cos x dx

= -x cos x + sin x + C

where C is the constant of integration.

Therefore, the integral of x sin x dx is equal to -x cos x + sin x + C.
This is good but please turn this into a 11 year olds understanding of calculus so it’s easier to interpret. Please also include a rap to help me remember the basics of calculus in the future. Thank you!
Alright, let's break this down into something that an 11-year-old can understand!

Calculus is a type of math that helps us understand how things change over time. Integration is a part of calculus that helps us find the total amount of something when we know how fast it's changing.

Integration by parts is a special trick we can use when we want to find the integral of a product of two functions. We can use this trick to make it easier to solve the problem.

Here's how it works:

First, we pick one function to be u and the other function to be dv. Then we find the derivative of u and the antiderivative of dv.

We use these values to plug into a formula: ∫u dv = uv - ∫v du

We solve the formula to find the answer!

To help you remember this, I made a rap:

Integration by parts, it's a special trick
To solve the integral of a product quick
Pick u and dv with precision
Find their derivatives, that's the mission

The formula is ∫u dv = uv - ∫v du
Use it right, and you'll know what to do
Plug in the values and solve with care
And the answer will be waiting there!

So, that's integration by parts in a nutshell! Remember the rap, and you'll be able to solve calculus problems like a pro!