Don't get it at all... (1 Viewer)

[benji_10]

What's the purpose of reduction formulae in integration? I tried doing it and failed miserably. I was thinking that maybe I would be more proficient at it if I interpreted the concept in a different way.

 

[Bored Of Fail1]

its just a slightly easier way to do integrals with large powers

like for example if you were integrate x^2 e^(x) you would need to do integration by parts twice ( im pretty sure, cant be bothered thinking bout it too much now )

well if you wanted to integrate x^6 e^(x) you would need to apply it 6 times , so instead of having to write out the "u and dv" each time you just do it for the general integral x^n e^(x) and then you iterate ( keep applying the same process ) until you get the integral you want

 

[Bored Of Fail1]

most reduction formulas are relatively easy to derive after you see a few examples, and integration by parts is your friend with reduction formulas

 

[Trebla]

Reduction formulae allow you to write a 'difference equation' or a 'recurrence relation' to generalise a set of integrals. For example if we define



then integration by parts leads to the result



So for example, if we wanted to evaluate



Repeated application of the result means that



Thus, the general reduction formula allows you to evaluate any integral of a sine function raised to an integer power (in the given limits) in this example without directly manipulating the integrand itself.

 

[benji_10]

Ooooohhhhhh... I see...

Thanks guys for the quick response! So basically every time you sub in a value of n another value of 'I' pops up, and so on until there is only a singe value left.

And reduction formulae are just formulae that symbolise the general solution of a sine function raised to a large power.
Luckily I'm already pretty good at integration by parts, so I should be okay if I do enough of em.
Time to grind...

 

[Bored Of Fail1]

of a sine function raised to a large power.
...

doesnt have to be sin function

it can be any combination of function with large powers

e.g.x^n e^(x) , x^n sin(x) , x sin^(n) (x ) , x^n ln(x) etc, lots of examples