int_(1)^(4)(x^(2)-x)d_(n)

evaluate int_(0)^(1)x^(2)(1-x)^(n)dx

The value of int_(0)^(1)e^(x^(2)-x)dx is (a) 1(c)>e^(-(1)/(4))(d)

The value of int_(0)^(oo)(dx)/(1+x^(4)) is (a) same as that of int_(0)^(oo)(x^(2)+1dx)/(1+x) (b) (pi)/(2sqrt(2))( c) same as that of int_(0)^(oo)(x^(2)+1dx)/(1+x^(4))(d)(pi)/(sqrt(2))

int_(2)^(4) (3x^(2)+1)/((x^(2)-1)^(3))dx = (lambda)/(n^(2)) where lambda, n in N and gcd(lambda,n) = 1 , then find the value of lambda + n

(01)int(x^(4)+x^(2)+1)d(x^(2))

The value of (^nC_(0))/(n)+(^nC_(1))/(n+1)+(^nC_(2))/(n+2)+....+(n)/(2n) is equal to a.int_(0)^(1)x^(n-1)(1-x)^(n)dxbint_(1)^(2)x^(n)(x-1)^(n-1)dxc*int_(1)^(2)x^(n-1)(1+x)^(n)dx d.int_(0)^(1)(1-x)^(n-1)dx

Evaluate: ( i ) int(x^(4)+x^(2)+1)d(x^(2)) (ii) int sin(e^(x))d(e^(x))

If m gt 0, n gt 0 , the definite integral l=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx depends upon the vlaues of m and n and is denoted by beta(m,n) , called the beta function. E.g. int_(0)^(1)x^(4)(1-x)^(5)dx=int_(0)^(1)x^(5-1)(1-x)^(6-1)dx=beta(5, 6) and int_(0)^(1)x^(5//2)(1-x)^(-1//2)dx=int_(0)^(1)x^(7//2-1)(1-x)^(1//2-1)dx=beta((7)/(2),(1)/(2)) . Obviously, beta(n, m)=beta(m, n) . The integral int_(0)^(pi//2)cos^(2m)theta sin^(2n) theta d theta is equal to

If m gt 0, n gt 0 , the definite integral l=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx depends upon the vlaues of m and n and is denoted by beta(m,n) , called the beta function. E.g. int_(0)^(1)x^(4)(1-x)^(5)dx=int_(0)^(1)x^(5-1)(1-x)^(6-1)dx=beta(5, 6) and int_(0)^(1)x^(5//2)(1-x)^(-1//2)dx=int_(0)^(1)x^(7//2-1)(1-x)^(1//2-1)dx=beta((7)/(2),(1)/(2)) . Obviously, beta(n, m)=beta(m, n) . If int_(0)^(n)(1-(x)/(n))^(n)x^(k-1)dx=R beta(k, n+1) , then R is equal to