If I(m,n)=int0^1x^(m-1)(1-x)^(n-1)dx, th...

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

`I(m,n)=int_(0)^(oo) (x^(m-1))/((1+x)^(m-n))dx`

`I(m,n)=int_(0)^(oo) (x^(m))/((1+x)^(m+n))dx`

`I(m,n)=int_(0)^(oo) (x^(n-1))/((1+x)^(m+n))dx`

`I(m,n)=int_(0)^(oo)(x^(n))/((1+x)^(m+n))dx`

Text Solution

Verified by Experts

The correct Answer is:

Putting `x-1/(1+y), dx=-1/((1+y)^(2))dy`, we get
`I(m,n)=int_(0)^(1) x^(m-1)(1-x)^(n-1)dx`
`=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`
Since `I(m,n)=I(n,m)` we have
`I(m,n)=int_(0)^(oo) (x^(m-1))/((1+x)^(m+n))dx=int_(0)^(oo)(x^(n-1))/((1+x)^(m+n))dx`

Topper's Solved these Questions

DEFINITE INTEGRATION
CENGAGE|Exercise Exercise (Multiple)|27 Videos
DEFINITE INTEGRATION
CENGAGE|Exercise Exercise (Comprehension)|31 Videos
DEFINITE INTEGRATION
CENGAGE|Exercise Exercise 8.11|6 Videos
CURVE TRACING
CENGAGE|Exercise Exercise|24 Videos
DETERMINANT
CENGAGE|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

If I(m,n)=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx, then

If I(m,n)=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx,(m,n in I,m,n>=0),th epsilonI(m,n)=int_(0)^(oo)(x^(m-1))/((1+x)^(m-n))dxI(m,n)=int_(0)^(oo)(x^(m-1))/((1+x)^(m+n))dxI(m,n)=int_(0)^(oo)(x^(n-1))/((1+x)^(m+n))dxI(m,n)=int_(0)^(oo)(x^(n))/((1+x)^(m+n))dx

If I(m,n)=int_(0)^(1)t^(m)(1+t)^(n).dt, then the expression for I(m,n) in terms of I(m+1,n-1) is:

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

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If mgt0, ngt0 , the definite integral I=int_0^1 x^(m-1)(1-x)^(n-1)dx depends upon the values of m and n is denoted by beta(m,n) , called the beta function.For example, int_0^1 x^3 (1-x)^4dx=int_0^1 x^(4-1) (1-x)^(5-1) dx=beta(4,5) and int_0^1 x^(3/2) (1-x)^((-1)/2)dx=int_0^1 x^(5/2-1) (1-x)^(1/2-1)dx=beta(5/2,1/2) .Obviously, beta(n,m)=beta(m,n) .Now answer the question:If int_0^oo x^(m-1)/(1+x)^(m+n)dx=k int_0^oo x^(n-1)/(1+x)^(m+n)dx , then k is equal to (A) m/n (B) 1 (C) n/m (D) none of these

CENGAGE-DEFINITE INTEGRATION -Exercise (Single)

lim(xto oo) (int(0)^(x)tan^(-1)dt)/(sqrt(x^(2)+1)) is equal to
Text Solution
|
A function f is defined by f(x) = int0^pi cos t cos(x-t)dt,0 <= x <= 2...
Text Solution
|
If f' is a differentiable function satisfying f(x)=int(0)^(x)sqrt(1-f^...
Text Solution
|
If int 0^1e^(x^2)(x-alpha)dx=0, then
Text Solution
|
The value of the integral int0^1e^x^2dx lies in the interval (0,1) ...
Text Solution
|
Given that f satisfies |f(u)-f(v)|lt=|u-v|forua n dv in [a , b]dot The...
Text Solution
|
The value of the integral int0^oo(xlogx)/((1+x^2)^2)dxi s 0 (b) log ...
Text Solution
|
int0^oo(pi/(1+pi^2x^2)-1/(1+x^2))logxdxi se q u a lto -pi/21npi (b) ...
Text Solution
|
If A=int0^pi cosx/(x+2)^2 \ dx, then int0^(pi//2) (sin 2x)/(x+1) \ dx ...
Text Solution
|
int(0)^(4)((y^(2)-4y+5)sin(y-2)dy)/([2y^(2)-8y+1]) is equal to
Text Solution
|
int(sin theta)^(cos theta) f(x tan theta)dx (where theta!=(npi)/2,n ep...
Text Solution
|
If I1=int0^1(e^x)/(1+x) dxaand I2=int0^1 x^2/(e^(x^3)(2-x^3)) dx then ...
Text Solution
|
L e tI1=int(-2)^2(x^6+3x^5+7x^4)/(x^4+2)dxa n d I2=int(-3)^1(2(x+1)^2...
Text Solution
|
Let f beintegrable over [0,a] for any real value of a. If I(1)=int(0...
Text Solution
|
The value of inta^b(x-a)^3(b-x)^4dxi s ((b-a)^4)/(6^4) (b) ((b-a)^8)...
Text Solution
|
If I(m,n)=int0^1x^(m-1)(1-x)^(n-1)dx, then
Text Solution
|
T h ev a l u eoft h ed efin i t ein t egr a lint0^(pi/2)(sin5x)/(sinx)...
Text Solution
|
If In=int0^pi e^x(sinx)^n dx , then (I3)/(I1) is equal to
Text Solution
|
If f'(x)=f(x)+int(0)^(1)f(x)dx,given f(0)=1, then the value of f(log(e...
Text Solution
|
Let f(x) be positive, continuous, and differentiable on the interval ...
Text Solution
|