Derivation and application of reduction formula?
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"Use integration by parts to derive the reduction formula \int\cos^n(x)dx=1/n\sinx\cos^(n-1)(x)+(n-1)/n\int\cos^(n-2)(x)dx , where n is a positive integer."
I assume that I need to split this into \int\cosx\cos^(n-1)x or something to get a valid IBP process?
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Use the previous reduction formula or integration by parts to evaluate: \int\cos^3dx
(Where is the x in this problem??)
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"Use integration by parts to derive the reduction formula
\int\cos^n(x)dx=1/n\sinx\cos^(n-1)(x)+(n-1)/n\int\cos^(n-2)(x)dx , wheren is a positive integer."
I assume that I need to split this into\int\cosx\cos^(n-1)x or something to get a valid IBP process? -
Use the previous reduction formula or integration by parts to evaluate:
\int\cos^3dx
(Where is thex in this problem??)
1 Answer
Write the integrand as:
then we can integrate by parts:
Now use:
and using the linearity of the integral:
The integral:
appears now on both sides of the equation and we can solve for it:
which proves the reduction formula.
For
and simplifying: