There is no universal technique known, which will solve any indefinite integral. There are certain techniques, such as substitution, partial fractions, integration by parts; but these apply to only certain types of integrals.
A reduction formula is another such technique in the form of recurrence relation.
A complicated integral is expressed using simpler integrals of the same kind. The integrals involve an integer parameter. The complicated integral is represented in terms of or , where is an integer. Ultimately, when or , or can be evaluated easily.
So, we reduce to a simpler form to get the answer.
Reduction Formulas for Trigonometric Functions
2 cases arise for such integrals, where only 1 integer parameter is there; can be odd or even.
When 2 integer parameters are there, and , we have 2 cases; when both and are even and otherwise.
Reduction Formulas for Product of Algebraic and Trigonometric Functions
In these problems, we will have functions of the kind etc. and trigonometric functions. We will use integration by parts and keep the first function algebraic. (i.e. ).
Reduction Formulas for Product of Exponential and Trigonometric Functions
In these problems, we will have functions of the kind etc. and trigonometric functions. We will use integration by parts and keep the first function trigonometric. (i.e. ).
Reduction Formulas for Product of Algebraic and Other Functions
In this case, we have to choose the first function to get the answer by integration by parts. There is no rule of thumb for the choice of first function. It totally depends on the functions in the integrand.
Useful Information for Definite Integrals
A definite integral has lower and upper limits. For example,
If the limits are such that , check whether is even or odd. If it is odd, then is . If it is even, then
Also, for any definite integral, following relation is true: