Prove that for `ngt1`. `int_0^1(cos^-1x)^ndx=n(pi/2)^(n-1)-n(n-1)int_0^1(cos^-1x)^(n-2)dx`

Evaluate: int_0^1x(1-x)^n dx

The value of ("^n C_0)/n + ("^nC_1)/(n+1) + ("^nC_2)/(n+2) +....+ ("^nC_ n)/(2n) is equal to a. int_0^1x^(n-1)(1-x)^n dx b. int_1^2x^n(x-1)^(n-1)dx c. int_1^2x^(n-1)(1+x)^n dx d. int_0^1(1-x)^(n-1)dx

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

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=

Q. int_0^pie^(cos^2x)( cos^3(2n+1) x dx, n in I

Prove that the value of : int_0^pi(sin(n+1/2)x)/sin(x/2)dx=pi

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=

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

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

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.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