Integration by reduction formulae
Part of a series of articles about  
Calculus  



Specialized 

Integration by reduction formula in integral calculus is a technique of integration, in the form of a recurrence relation. It is used when an expression containing an integer parameter, usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree, can't be integrated directly. But using other methods of integration a reduction formula can be set up to obtain the integral of the same or similar expression with a lower integer parameter, progressively simplifying the integral until it can be evaluated. ^{[1]} This method of integration is one of the earliest used.
How to find the reduction formula
The reduction formula can be derived using any of the common methods of integration, like integration by substitution, integration by parts, integration by trigonometric substitution, integration by partial fractions, etc. The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by I_{n}, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example I_{n1} or I_{n2}. This makes the reduction formula a type of recurrence relation. In other words, the reduction formula expresses the integral
in terms of
where
How to compute the integral
To compute the integral, we set n to its value and use the reduction formula to calculate the (n – 1) or (n – 2) integral. The higher index integral can be used to calculate lower index ones; the process is continued repeatedly until we reach a point where the function to be integrated can be computed, usually when its index is 0 or 1. Then we backsubstitute the previous results until we have computed I_{n}. ^{[2]}
Examples
Below are examples of the procedure.
Cosine integral
Typically, integrals like
can be evaluated by a reduction formula.
Start by setting:
Now rewrite as:
Integrating by this substitution:
Now integrating by parts:
solving for I_{n}:
so the reduction formula is:
To supplement the example, the above can be used to evaluate the integral for (say) n = 5;
Calculating lower indices:
backsubstituting:
where C is a constant.
Exponential integral
Another typical example is:
 .
Start by setting:
Integrating by substitution:
Now integrating by parts:
shifting indices back by 1 (so n + 1 → n, n → n – 1):
solving for In:
so the reduction formula is:
Tables of integral reduction formulas
Rational functions
The following integrals^{[3]} contain:
 Factors of the linear radical
 Linear factors and the linear radical
 Quadratic factors
 Quadratic factors , for
 Quadratic factors , for
 (Irreducible) quadratic factors
 Radicals of irreducible quadratic factors
Integral  Reduction formula 

Integral  Reduction formula 

 

Integral  Reduction formula 

Integral  Reduction formula 

Integral  Reduction formula 

Integral  Reduction formula 

Integral  Reduction formula 

note that by the laws of indices:
Transcendental functions
See main article: Transcendental function
The following integrals^{[4]} contain:
 Factors of sine
 Factors of cosine
 Factors of sine and cosine products and quotients
 Products/quotients of exponential factors and powers of x
 Products of exponential and sine/cosine factors
Integral  Reduction formula 


the formulae can be combined to obtain separate equations in I_{n}:
and J_{n}:

Integral  Reduction formula 

Integral  Reduction formula 





References
 ↑ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 9780521861533
 ↑ Further Elementary Analysis, R.I. Porter, G. Bell & Sons Ltd, 1978, ISBN 0713515945
 ↑ http://www.sosmath.com/tables/tables.html > Indefinite integrals list
 ↑ http://www.sosmath.com/tables/tables.html > Indefinite integrals list
Bibliography
Wikibooks has a book on the topic of: Calculus/Integration techniques/Reduction Formula 
 Anton, Bivens, Davis, Calculus, 7th edition.