Integration by reduction formulae
In integral calculus, integration by reduction formulae is method relying on recurrence relations. 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. 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, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example "I" or "I". 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 express it in terms of the ("n" – 1) or ("n" – 2) integral. The lower index integral can be used to calculate the higher 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 back-substitute the previous results until we have computed "I". Examples. Below are examples of the procedure. Cosine integral Typically, integrals like can be evaluated  by a reduction formula. Start by setting: Now re-write as: Integrating by this substitution: Now integrating by parts: solving for "I": 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: back-substituting: 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 "I": so the reduction formula is: An alternative way in which the  derivation could be done starts by substituting formula_29. Integration by substitution: formula_30 formula_31 Now integrating by parts: formula_32 which gives the reduction formula when substituting back: formula_27 which is equivalent to: Another alternative way in which the derivation could be done by integrating by parts: Remember: which gives the reduction formula when substituting back: which is equivalent to: Tables of integral reduction formulas. Rational functions. The following integrals contain: note that by the laws of indices: Transcendental functions. The following integrals contain: