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Chapter: 12th Maths : UNIT 9 : Applications of Integration

Gamma Integral

Applications of Integration | Mathematics

Gamma Integral

In this section, we study about a special improper integral of the form 0 e x xn1dx , 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 ex xn dx = n!, where n is a positive integer.

Solution

Applying integration by parts, we get


Let In = 0 ex xn dx .Then, In = nIn−1 .

So, we get In = n ( n1)In2 .

Proceeding in this way, we get ultimately,

In = n ( n1)( n2) ( 2)(1)I0 .

But, I0 = 0 e x x0dx = ( ex)0 = 0 +1 = 1 . So, we get In = n (n 1)(n 2) (2)(1) = n!.

Hence, we get


Result

0 e x xn dx = n!, where n is a nonnegative integer.

Note

The integral 0 ex xn1dx defines a unique positive integer for every positive integer n 1.


Definition 9.1

0 e x xn1dx 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 xn 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




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