Evaluate: `inte^x(f(x)+f^(prime)(x))dx=e^xf(x)+C`

Evaluate: int[f(x)g^(x)-f^(x)g(x)]dx

Evaluate int[f(x)g^(n)(x)-f^(n)(x)g(x)]dx

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

Evaluate int_(0)^(a)(f(x))/(f(x)+f(a-x)) dx.

Statement 1: int({f(x)varphi^(prime)(x)-f^(prime)(x)varphi(x)})/(f(x)varphi(x)) -logf(x)dx=1/2{(varphi(x))/(f(x))}^2+c Statement 2 : int(h(x))^n h^(prime)(x)dx=((h(x))^(n+1))/(n+1)+c

If function f satisfies the relation f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) for all x ,

If function f satisfies the relation f(x)xf^(prime)(-x)=f(-x)xf^(prime)(x)fora l lx ,a n df(0)=3,a n diff(3)=3, then the value of f(-3) is ______________

Suppose fa n dg are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all xa n df^(prime)a n dg' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x))e q u a l (a) (-2f^(prime)(x))/f (b) (2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))

Using the first principle, prove that d/(dx)(1/(f(x)))=(-f^(prime)(x))/([f(x)]^2)