Determine a positive integer `nlt=5` such that `int_0^1e^x(x-1)^n=16-6e`

Determine a positive integer n such that int_0^(pi/2)x^nsinx dx=3/4(pi^2-8)

f(x)=e^(-1/x),w h e r ex >0, Let for each positive integer n ,P_n be the polynomial such that (d^nf(x))/(dx^n)=P_n(1/x)e^(-1/x) for all x > 0. Show that P_(n+1)(x)=x^2[P_n(x)-d/(dx)P_n(x)]

Evaluate: int_0^1 e^(2-3x) dx

Evaluate int_0^1(e^x)/(1+e^(2x)) dx

Let m and n be two positive integers greater than 1.If lim_(alpha->0) (e^(cos alpha^n)-e)/(alpha^m)=-(e/2) then the value of m/n is

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)^(oo)(x^(m-1))/((1+x)^(m+n))dx=

int_(0)^(1)e^(2x)e^(e^(x) dx =)

Evaluate int_(0)^(1) x e^(-2x) dx