reduction formulaescript for

Question

 

Reduction formulae

Script for Introduction:

First let us know what is meant by reduction formula.

In simple words,

                A formula which expressess(or reduces) the integral of the n^th  indexed function in terms of that of n-1^th  indexed function is called a Reduction formula.

 

Let us see how reduction formula helps us in solving the problems in different cases:

 

Problem 1:

Solve  Integration of

sin⁴x dx

Script for this problem:

This problem can be solved by reduction formula for sine which states that,

// print line 1

 

Here we have n=4.

Therefore substituting the value  of n=4 in the given problem,we have,

//print line 2

 

Which is equal to,

//print line 3

 

On expanding  I₂ in a similar manner, we can write I₄ as,

//print line 4

 

Now, we know that,

//print line5

 

Which is equal to,

//print line 6

 

Here substituting the value of I₀ in the equation of I₄ ,we have

//print line 7

 

Thus, by reduction formula  the given problem is solved.