int x^(n)e^(-x)dx=-x^(n)e^(-x)+n int x^(n-1)e^(-x)dx

If I_(n) = int x^(n) e^(-x) dx prove that I_(n) = -x^(n) e^(-x) + nI_(n-1)

int_(n)^(n+1)f(x)dx=n^(2)+n then int_(-1)^(1)f(x)dx=

underset is If int_(0)^(oo)e^(-ax)dx=(1)/(a), then int_(0)^(oo)(x^(n))e^(-ax)dx

int e^(x)/(e^(x)+1) dx =

int (dx)/(e^(x)+e^(-x)) dx =

int(1)/(e^(x)+e^(-x))dx

int (e^(x))/(1+e^(x))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-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=

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=