Integration by Parts

Integration by parts is a useful strategy for simplifying some integrals. It is based on the combination rule for differentiation and the general approach can be summarized by:

This technique is particularly appropriate for removing a linear term multiplying an exponential. For example, the integral

can be simplified by the identification

Then u can be differentiated and dv can be integrated.

The value of the integral is then

Average particle energy
Mean particle lifetime

If a higher power term of x multiplies the exponential, then the process of integration by parts can be repeated to reduce the power term. Two stages are necessary to integrate a square term.

Hydrogen ground state probability

Three stages are necessary if the integrand has a cubic term, and so on.

Radius expectation value, hydrogen atom

Integration concepts
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