Reduction Formula (1 Viewer)

[someth1ng]

I can't seem to do any reduction formulae question, it just doesn't make any sense to me...

I need help with these:
Integral x^n (1+x^2)^0.5
Integral sin^5 (x)
Integral cos^6 (x)
Integral e^4n (1-x)^3

I just don't understand what the aim of this is, I know it's to make the powers smaller but I don't understand why it's needed or how to do it. What exactly does it mean by say I(n-1) or I(n-2) etc?

Help appreciated,
I find Integration harder than Polynomials, Volumes, Complex and Graphs...=/

 

[Shadowdude]

Why you need to do it? Well, cos(x)^6 and sin(x)^5 can't be easily solved directly. So you use a reduction formula to reduce the powers so in the end you just have to integrate a sin or a cos or something easy like that.

 

[nightweaver066]

e.g.







Your first question,



















Notice how the degree of the integral on the RHS reduces each time?

So if you're integrating something like , you simply sub n = 6 in to that equation, and keep reusing that same formula til you end up with the integral.

 

[juantheron]

 

[juantheron]

 

[juantheron]

 

[juantheron]

 

[someth1ng]

So essentially, reduction formulas are doing anything to reduce the magnitude of the powers?

 

[Carrotsticks]

So essentially, reduction formulas are doing anything to reduce the magnitude of the powers?

Exactly, hence the name.

ie: We can't integrate sin^7(x) directly very easily, so we reduce the power to sin(x) or maybe even sin^0(x), which is = 1, then we integrate it since it is familiar.

 

[Inference]

Reduction formulas are based on recurrence relationships. What you need to realise is that we are essentially computing some integral by continually reducing it down into an integral, where . To find the integral, we first set n to its initial value while then reducing it down to lower indices until we can actually compute the expression with the lower index. Now initially, you may need a bit of wishful thinking to get used to doing these, however after a while you'll start to pick up tricks as with any integration technique.