If `f(x)` is polynomaial function of degree n, prove that `int e^x f(x) dx=e^x[f(x)-f'(x)+f''(x)-f'''(x)+......+(-1)^n f^n (x)]` where `f^n(x)=(d^nf)/(dx^n)`

Evaluate: int(x)i spol y nom i a lfu n c t ionoft h en t h degr e e ,p rov et h a t- inte^xf(x)dx=e^x[f(x)f^(prime)(x)+f^(x)=f^(x)++(-1)^nf^((n))(x)] Where f^((n))(x)d e not e s(d^nf)/(dx^n)

If f(x) is a polynomial of degree n such that f(0)=0 , f(x)=1/2,....,f(n)=n/(n+1) , ,then the value of f (n+ 1) is

int e^x {f(x)-f'(x)}dx= phi(x) , then int e^x f(x) dx is

If (d(f(x)))/(dx) = e^(-x) f(x) + e^(x) f(-x) , then f(x) is, (given f(0) = 0)

If f(x) = (a-x^n)^(1/n), a > 0 and n in N , then prove that f(f(x)) = x for all x.

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

Statement-1: int_(0)^(npi+v)|sin x|dx=2n+1-cos v where n in N and 0 le v lt pi . Stetement-2: If f(x) is a periodic function with period T, then (i) int_(0)^(nT) f(x)dx=n int_(0)^(T) f(x)dx , where n in N and (ii) int_(nT)^(nT+a) f(x)dx=int_(0)^(a) f(x) dx , where n in N

If f(x) is a polynomial of degree n(gt2) and f(x)=f(alpha-x) , (where alpha is a fixed real number ), then the degree of f'(x) is

If f(x), g(x), h(x) are polynomials of three degree, then phi(x)=|(f'(x),g'(x),h'(x)), (f''(x),g''(x),h''(x)), (f'''(x),g'''(x),h'''(x))| is a polynomial of degree (where f^n (x) represents nth derivative of f(x))

If int f(x)dx = F(x), f(x) is a continuous function,then int (f(x))/(F(x))dx equals