TeXercises.com

Radial Integrals

About points...

We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.

That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.

About difficulty...

We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.

That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:

Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise

Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise

Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise

Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise

Level 5 -
Level 6 -

Exercise

https://texercises.com/exercise/radial-integrals/

Question

Solution

Short

Video

\(\LaTeX\)

No explanation / solution video to this exercise has yet been created.

Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!

Exercise:
Verify the following egrals: abcliste abc I_n _^infty r^n e^-r/a textrmdr n! a^n+ abc I'_n _^infty r^n lefte^-r/a right^ textrmdr fracn! a^n+^n+ abcliste

Solution:
abcliste abc For n the egral is I_ _^infty r^ e^-r/a textrmdr _^infty e^-r/a textrmdr -a e^-r/a Big|_^infty Since e^-r/ato for rtoinfty it follows I_ + a e^/a a For n we use egration by parts with frr^n and g're^-r/a: I_n leftr^n-ae^-r/aright_^infty-_^infty n r^n--a e^-r/a textrmdr Since for any n r^n e^-r/a &to quad textrmfor rtoinfty and ^n e^-/a times the first term vanishes. It follows that I_n n a _^infty r^n- e^-r/a textrmdr n a I_n- Continuing recursively the egral can be written as I_n n a times n- a times n- a times s times a times I_ n! a^n times a n! a^n+ quad square abc Since lefte^-r/aright^ e^-r/a e^-r/a' with a'a/ we can write the egral I'_n as I'_n _^infty r^n e^-r/a' textrmdr n! a'^n+ n! leftfracaright^n+ fracn! a^n+^n+ quad square abcliste

Meta Information

\(\LaTeX\)-Code

Contained in these collections:

Hydrogen Atom by by

15 | 15

Attributes & Decorations

Branches	Integrals
Tags	integration by parts, radial, spherical coordinates
Content image
Difficulty	(2, default)
Points	0 (default)
Language	(English)
Type	Calculative / Quantity
Creator	by
Decoration
File
Link