Integration by parts is a technique used in calculus to solve certain integrals, particularly those involving products of functions. It is based on the product rule for differentiation and can be thought of as the "inverse" of the product rule. The formula for integration by parts is derived from the product rule and is given by:

∫u dv = uv - ∫v du

where u and v are functions of x, and du and dv are their respective derivatives with respect to x.

To use integration by parts, follow these steps:

1. Choose u and dv from the integrand (the product of functions you're trying to integrate). Typically, choose u as a function that becomes simpler when differentiated and dv as a function that can be easily integrated.
2. Differentiate u to find du, and integrate dv to find v.
3. Plug u, v, and du into the integration by parts formula.
4. Evaluate the remaining integral, if possible, and simplify the result.

Here's an example to illustrate the process:

Find the integral of x * e^x dx:

1. Choose u and dv:
u = x, dv = e^x dx

2. Differentiate u to find du, and integrate dv to find v:
du = dx, v = e^x

3. Plug u, v, and du into the integration by parts formula:
∫x * e^x dx = x * e^x - ∫e^x dx

4. Evaluate the remaining integral and simplify the result:
∫x * e^x dx = x * e^x - e^x + C

In this example, C represents the constant of integration. The final result is:

∫x * e^x dx = (x - 1) * e^x + C
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