Certain definite integrals can be evaluated by an index-reduction method.

**Reduction Formulae**

Certain
definite integrals can be evaluated by an index-reduction method. In this
section, we obtain the values of the following definite integrals:

We also
obtain the value of the improper integral ^{âˆž}âˆ«_{0} *e*^{âˆ’}^{x}*x ^{n}*

The
method of obtaining a reduction formula has the following steps:

**Step 1 :** Identify an index (positive
integer) *n* in the integral.

**Step 2 :** Put the integral as I_{n}.

**Step 3 :** Applying integration by parts,
obtain the equation for I_{n} in terms of I_{nâˆ’1} or I_{nâˆ’2}.

The
resulting equation is called the reduction formula for I_{n}.

We list
below a few reduction formulae without proof:

Using
the reduction formulas I and II, we obtain the following result (stated without
proofs):

By
applying the reduction formula III iteratively, we get the following results
(stated without proof):

(i) If *n* is even and *m* is even,

(ii) If *n* is odd and *m* is any positive integer (even or odd), then

**Note**

If one
of *m* and *n* is odd, then it is convenient to get the power of cos *x* as odd. For instance, if *m *is odd and* n *is even, then

** **

**Example 9.39**

Find the
values of the following:

By
applying the reduction formula III iteratively, we get the following results
(stated without proof):

Tags : Applications of Integration | Mathematics , 12th Maths : UNIT 9 : Applications of Integration

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12th Maths : UNIT 9 : Applications of Integration : Reduction Formulae | Applications of Integration | Mathematics

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