Home
Class 12
MATHS
Let m,n be two positive real numbers and...

Let m,n be two positive real numbers and define `f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx` and `g(m,n)=int_(0)^(1)x^(m-1)(1-m)^(n-1)dx`.
It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0.
`int_(0)^(1)x^(m)(log_(e).(1)/(x))dx=`

A

g(m,n)

B

`g(m-1,n)`

C

`g(m-1,n-1)`

D

`g(m,n-1)`

Text Solution

Verified by Experts

The correct Answer is:
A
`g(m,n)=int_(0)^(1)x^(m-1)(1-x)^(n-1)dt`
Put `x=(1)/(1+y)`
`rArr" "g(m,n)=int_(oo)^(0)(1)/((1+y)^(m-1))(1-(1)/(1+y))^(n-1)(-(1)/((1+y)^(2)))dy`
`" "=int_(0)^(oo)(y^(n-1))/((1+y)^(m+n))dy`
`" "=int_(0)^(oo)(x^(n-1))/((1+x)^(m+n))dx`
Promotional Banner

Similar Questions

Explore conceptually related problems

Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx . It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0 . int_(0)^(1)(x^(m-1)+x^(n-1))/((1+x)^(m+n))dx=

If I(m,n)=int_0^1x^(m-1)(1-x)^(n-1)dx , then

Evaluate : int_(0)^(1)x(1-x)^(n)dx

If I_(m)=int_(0)^(oo) e^(-x)x^(n-1)dx, "then" int_(0)^(oo) e^(-lambdax) x^(n-1)dx

Evaluate int_(0)^(1)((log(1/x))^(n-1)dx .

Let I_(n) = int_(0)^(1)x^(n)(tan^(1)x)dx, n in N , then

Let I_(n) = int_(0)^(1)(1-x^(3))^(n)dx, (nin N) then

If I_(m,n)= int_(0)^(1) x^(m) (ln x)^(n) dx then I_(m,n) is also equal to

If = int_(0)^(1) x^(n)e^(-x)dx "for" n in N "then" I_(n)-nI_(n-1)=

The value of the integral int_(0)^(1) x(1-x)^(n)dx is -