Ugh I hate math, integration by parts doesn't make any sense

" i guess it 's simple math . you do n't have to be a mathematician for math to be the math . " " it does n't matter to me . it just helps me to concentrate . i 'm learning a great deal . " " you are learning very fast . " " yeah . i have a lot to learn from you . " " i think you have more than you think you have . " " i hope you 're right . " he nodded and they sat in silence for a few more minutes . he was n't sure what to say . his mind had been racing . he felt like he was in the middle of some kind of crazy dream and he was having an out - of - body experience . " i 'm glad you 're here , " he said . " me too . " she smiled .

And things start to get really stupid when you do calculations with more than one variable.

I'd like to see a list of example functions, and then I'd like to see how the functions are used.

I think that's a pretty reasonable request.

Don't worry, many people feel the same way about math. Integration by parts can be confusing at first, but with a little practice it becomes easier to understand. Remember, there is no one right way to do integration by parts. If you don't understand how to do it one way, try another. The most important thing is to practice and get comfortable with the process.

I understand that integration by parts can be tricky to grasp at first. I'll do my best to help simplify the concept for you.

Integration by parts is a technique used to integrate a product of two functions. It's based on the product rule for differentiation, which states that the derivative of a product of two functions is equal to the sum of: the first function times the derivative of the second function, and the second function times the derivative of the first function.

Mathematically, the product rule is expressed as:

(d(uv)/dx) = u(dv/dx) + v(du/dx)

When we integrate both sides of this equation with respect to x, we get:

∫ (d(uv)/dx) dx = ∫ u(dv/dx) dx + ∫ v(du/dx) dx

The left side simplifies to the product uv, since the integral of a derivative cancels the derivative, leaving us with the original function:

uv = ∫ u(dv/dx) dx + ∫ v(du/dx) dx

Now, if you want to integrate the product of u and a function derived from v (dv/dx), you can rewrite the above formula as:

∫ u(dv/dx) dx = uv - ∫ v(du/dx) dx

This is the formula for integration by parts.

Here's a step-by-step guide to using integration by parts:

1. Choose which function will be u and which will be dv/dx (or, equivalently, which will be dv). This choice can often impact the ease of the resulting integration.

2. Differentiate u with respect to x to find du/dx (or, equivalently, find du).

3. Integrate dv/dx with respect to x to find v.

4. Plug u, v, and du/dx (or, equivalently, du) into the integration by parts formula:

   ∫ u(dv/dx) dx = uv - ∫ v(du/dx) dx

5. Evaluate the right-hand side of the equation. If needed, you may have to use integration by parts again.

It may take some practice to get comfortable with this technique, but I'm here to help. If you'd like, you can provide an example problem, and we can work through it together.

You're not alone—integration by parts is one of those topics that feels like wizardry at first. But it’s really just the reverse of the product rule for derivatives.

Let me break it down in a way that might make it click:

The Idea Behind Integration by Parts

The product rule for derivatives says:

d/dx (u v) = u′v + u v′

If we integrate both sides:

∫ u′v dx + ∫ u v′ dx = u v

Rearranging gives:

∫ u dv = u v − ∫ v du

That’s all integration by parts is: undoing the product rule.

The Formula

∫ u dv = u v − ∫ v du

You just need to choose:

• (u): the part that’s easy to differentiate
• (dv): the part that’s easy to integrate

Example: (∫ x e^x dx)

1. Pick (u = x) (because differentiating (x) gives (1))

   Then (du = dx).

2. The rest is (dv = e^x dx)

   So (v = ∫ e^x dx = e^x).

3. Apply the formula:

   ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C

Would you like me to give you a “cheat sheet” for choosing (u) and (dv) so it’s less guesswork?