Show that `inte^x(f(x)+f^(prime)(x))dx=e^xf(x)+C`

Let inte^x{f(x)-f^(prime)(x)}dx=varphi(x)dot Then inte^xf(x)dx is

Statement-I: int e^(x) sinxdx=(e^(x))/(2)(sinx-cosx)+c Statement-II: int e^(x)(f(x)+f'(x))dx=e^(x)f(x)+c

Show that int_(0)^(a)f(x)g(x)dx=2int_(0)^(a)f(x)dx , if f and g are defined as f (x) = f (a - x) and g(x) + g(a - x) = 4

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)

If intg(x)dx=g(x) , then evaluate intg(x){f(x)+f^(prime)(x)}dx

Solve x dy=(y+x(f(y/x))/(f^(prime)(y/x)))dx

If f(x)is integrable function in the interval [-a,a] then show that int_(-a)^(a)f(x)dx=int_(0)^(a)[f(x)+f(-x)]dx.

Suppose f(x)=e^(a x)+e^(b x) , where a!=b , and that f^(prime_ prime)(x)-2f^(prime)(x)-15f(x)=0 for all xdot Then the value of (|a b|)/3 is ___

Suppose f and g are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all x and f^(prime) and g' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x)) is equal (a) (-2f^(prime)(x))/f (b) (-2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))

A function y=f(x) satisfies (x+1)f^(prime)(x)-2(x^2+x)f(x)=((e^x)^2)/((x+1)),AAx >-1. If f(0)=5, then f(x) is