**Gamma Integral**

In this
section, we study about a special improper integral of the form ^{∞}∫_{0} *e*
^{−}^{x} *x*^{n}^{−}^{1}*dx* , where *n* is a positive integer. Here, we have

By
L’Hoˆpital’s rule, for every positive integer *m* , we get,

** **

**Example 9.43**

Prove
that ^{∞}∫_{0} *e*^{−}^{x} *x*^{n}
*dx* = *n*!, where *n* is a positive integer.

**Solution**

Applying
integration by parts, we get

Let In =
^{∞}∫_{0} e^{−x}* *x^{n} *dx* .Then, I_{n} = *n*I_{n−1}
.

So, we
get *I*_{n }= *n* ( *n* −1)*I*_{n}_{−}_{2} ._{}

Proceeding
in this way, we get ultimately,

*I*_{n} = *n* ( *n* − 1)( *n* − 2) ( 2)(1)*I*_{0} .

But, *I*_{0} =
^{∞}∫_{0} *e*
^{−}^{x} *x*^{0}*dx* = ( −*e*^{−}^{x})^{∞}_{0} = 0 +1 = 1 . So, we get *I*_{n} = *n*
(*n* − 1)(*n* − 2) (2)(1) = *n*!.

Hence,
we get

**Result**

^{∞}∫_{0}* e *^{−}^{x}* x*^{n}
dx =*
n*!, where* n *is a nonnegative
integer.

**Note**

The
integral ^{∞}∫_{0} *e*^{−}^{x} *x*^{n}^{−}^{1}*dx* defines a unique positive integer
for every positive integer *n* ≥
1.

**Definition 9.1**

^{∞}∫_{0}* e *^{−}^{x}* x*^{n}^{−}^{1}*dx *is called the **gamma integral**. It is denoted by Γ(*n*) and
is read as “gamma of *n* ”.

**Note**

**Example 9.44**

Evaluate
^{∞}∫_{0} *e*
^{−}^{ax} *x*^{n}
*dx* ,where *a* > 0 .

**Solution**

Making
the substitution *t* =
*ax* , we get *dt* = *adx*
and *x* = 0 ⇒ *t*
=
0 and *x* = ∞
⇒ *t*
= ∞
.

Hence, we
get

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

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12th Maths : UNIT 9 : Applications of Integration : Gamma Integral | Applications of Integration | Mathematics