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`

Evaluate: 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 varphi(x)= e^xf(x)

Let g^(prime)(x)>0a n df^(prime)(x) g(f(x-1)) f(g(x+1))>f(g(x-1)) g(f(x+1))

Evaluate: ifintg(x)dx=g(x),t h e nintg(x){f(x)+f^(prime)(x)}dx

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

Statement 1: For f(x)=sinx ,f^(prime)(pi)=f^(prime)(3pi)dot Statement 2: For f(x)=sinx ,f(pi)=f(3pi)dot

If int(dx)/(x^(2)+ax+1)=f(g(x))+c, then

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))

The general solution of the differential equation, y^(prime)+yvarphi^(prime)(x)-varphi^(prime)(x) varphi(x)=0 , where varphi(x) is a known function, is