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Reduction Formulas for Integrals

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illustrate with fixed n
choose n :
n
1
2
3
4
5
integrand:
x
n
x
n
x
sin(x)
n
x
cos(x)
n
sin
(x)
n
tan
(x)
n
log
(x)
n
x
x+1
To establish a reduction formula for ∫
x
n
x
dx,use integration by parts:
u =
n
x
dv =
x
dx
du = n
n-1
x
dx
v =
x
∫
x
n
x
dx =
x
n
x
- n∫
x
n-1
x
dx
The use of reduction formulas is one of the standard techniques of integration taught in a first-year calculus course. This Demonstration shows how
u
substitution, integration by parts, and algebraic manipulation can be used to derive a variety of reduction formulas. Selecting the "illustrate with fixed
n
" box lets you see how the reduction formulas are used for small values of
n
and shows more detail for the algebraic manipulations needed for some of the examples.
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