Reduction Formulas for Integrals

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.

Details

When "illustrate with fixed

n

" is selected and the integrand is

n

sin

(x)

or

n

tan

(x)

, the value

n=1

is not available. In those two cases, no further reduction is needed.

External Links

Integration by Parts (Wolfram MathWorld)

Permanent Citation

Marc Brodie

"Reduction Formulas for Integrals"
http://demonstrations.wolfram.com/ReductionFormulasForIntegrals/
Wolfram Demonstrations Project
Published: January 24, 2012